# Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde # This file is part of Math-PlanePath. # # Math-PlanePath is free software; you can redistribute it and/or modify # it under the terms of the GNU General Public License as published by the # Free Software Foundation; either version 3, or (at your option) any later # version. # # Math-PlanePath is distributed in the hope that it will be useful, but # WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY # or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License # for more details. # # You should have received a copy of the GNU General Public License along # with Math-PlanePath. If not, see . # Maybe: # "Turn" N of turn positions # $path->n_next_turn($n) package Math::NumSeq::PlanePathN; use 5.004; use strict; use Carp 'croak'; use constant 1.02; use vars '$VERSION','@ISA'; $VERSION = 129; use Math::NumSeq; @ISA = ('Math::NumSeq'); use Math::NumSeq::PlanePathCoord; # uncomment this to run the ### lines # use Smart::Comments; sub description { my ($self) = @_; if (ref $self) { return "N values on $self->{'line_type'} of path $self->{'planepath'}"; } else { # class method return 'N values from a PlanePath'; } } use constant::defer parameter_info_array => sub { return [ Math::NumSeq::PlanePathCoord::_parameter_info_planepath(), { name => 'line_type', display => 'Line Type', type => 'enum', default => 'X_axis', choices => ['X_axis', 'Y_axis', 'X_neg', 'Y_neg', 'Diagonal', 'Diagonal_NW', 'Diagonal_SW', 'Diagonal_SE', 'Depth_start', 'Depth_end', ], description => 'The axis or line to take path N values from.', }, ]; }; #------------------------------------------------------------------------------ my %oeis_anum = ( # MultipleRings,step=0 -- integers 1,2,3, etc, but starting i=0 ); sub oeis_anum { my ($self) = @_; ### PlanePathN oeis_anum() ... my $planepath_object = $self->{'planepath_object'}; { my $key = Math::NumSeq::PlanePathCoord::_planepath_oeis_anum_key($self->{'planepath_object'}); my $i_start = $self->i_start; if ($i_start != $self->default_i_start) { ### $i_start ### cf n_start: $planepath_object->n_start $key .= ",i_start=$i_start"; } ### planepath: ref $planepath_object ### $key ### whole table: $planepath_object->_NumSeq_N_oeis_anum ### key href: $planepath_object->_NumSeq_N_oeis_anum->{$key} if (my $anum = $planepath_object->_NumSeq_N_oeis_anum->{$key}->{$self->{'line_type'}}) { return $anum; } if (my $anum = $planepath_object->_NumSeq_N_oeis_all_anum->{$self->{'line_type'}}) { return $anum; } } { my $key = Math::NumSeq::PlanePathCoord::_planepath_oeis_key($planepath_object); my $i_start = $self->i_start; if ($i_start != $self->default_i_start) { ### $i_start ### cf n_start: $planepath_object->n_start $key .= ",i_start=$i_start"; } ### $key ### hash: $oeis_anum{$key} return $oeis_anum{$key}->{$self->{'line_type'}}; } } #------------------------------------------------------------------------------ sub default_i_start { my ($self) = @_; my $planepath_object = $self->{'planepath_object'} # nasty hack allow no 'planepath_object' when SUPER::new() calls rewind() || return 0; my $method = "_NumSeq_$self->{'line_type'}_i_start"; if (my $func = $planepath_object->can($method)) { return $planepath_object->$func(); } return 0; } sub new { my $self = shift->SUPER::new(@_); my $planepath_object = ($self->{'planepath_object'} ||= Math::NumSeq::PlanePathCoord::_planepath_name_to_object($self->{'planepath'})); ### $planepath_object my $line_type = $self->{'line_type'}; ### i_func name: "i_func_$line_type" $self->{'i_func'} = $self->can("i_func_$line_type") || croak "Unrecognised line_type: ",$line_type; $self->{'pred_func'} = $self->can("pred_func_$line_type") || croak "Unrecognised line_type: ",$line_type; if (my $func = $planepath_object->can("_NumSeq_${line_type}_step")) { $self->{'i_step'} = $planepath_object->$func(); } elsif ($planepath_object->_NumSeq_A2() && ($line_type eq 'X_axis' || $line_type eq 'Y_axis' || $line_type eq 'X_neg' || $line_type eq 'Y_neg')) { $self->{'i_step'} = 2; } else { $self->{'i_step'} = 1; } ### i_step: $self->{'i_step'} # for use in pred() $self->{'i_start'} = $self->i_start; $self->rewind; return $self; } sub rewind { my ($self) = @_; $self->{'i'} = $self->i_start; } sub next { my ($self) = @_; ### NumSeq-PlanePathN next(): "i=$self->{'i'}" my $i = $self->{'i'}; my $n = &{$self->{'i_func'}} ($self, $i); ### $n if (! defined $n) { ### i_func returns undef, no value ... return; } # secret experimental automatic bigint to preserve precision if (! ref $n && $n > 0xFF_FFFF) { $n = &{$self->{'i_func'}}($self,_to_bigint($i)); } return ($self->{'i'}++, $n); } sub _to_bigint { my ($n) = @_; # stringize to avoid UV->BigInt bug in Math::BigInt::GMP version 1.37 return _bigint()->new("$n"); } # or maybe check for new enough for uv->mpz fix use constant::defer _bigint => sub { # no choice of back-end here, leave that to the mainline require Math::BigInt; return 'Math::BigInt'; }; sub ith { my ($self, $i) = @_; ### NumSeq-PlanePathN ith(): $i return &{$self->{'i_func'}}($self, $i); } sub i_func_X_axis { my ($self, $i) = @_; my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n ($i * $self->{'i_step'}, $path_object->_NumSeq_X_axis_at_Y); } sub i_func_Y_axis { my ($self, $i) = @_; ### i_func_Y_axis(): "i=$i" ### X: $self->{'planepath_object'}->_NumSeq_Y_axis_at_X ### Y: $i * $self->{'i_step'} my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n ($path_object->_NumSeq_Y_axis_at_X, $i * $self->{'i_step'}); } sub i_func_X_neg { my ($self, $i) = @_; ### i_func_X_neg(): $i my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n (-$i * $self->{'i_step'}, $path_object->_NumSeq_X_axis_at_Y); } sub i_func_Y_neg { my ($self, $i) = @_; my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n ($path_object->_NumSeq_Y_axis_at_X, - $i * $self->{'i_step'}); } sub i_func_Diagonal { my ($self, $i) = @_; my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n ($i + $path_object->_NumSeq_Diagonal_X_offset, $i); } sub i_func_Diagonal_NW { my ($self, $i) = @_; my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n (-$i + $path_object->_NumSeq_Diagonal_X_offset, $i); } sub i_func_Diagonal_SW { my ($self, $i) = @_; my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n (-$i + $path_object->_NumSeq_Diagonal_X_offset, -$i); } sub i_func_Diagonal_SE { my ($self, $i) = @_; my $path_object = $self->{'planepath_object'}; return $path_object->xy_to_n ($i + $path_object->_NumSeq_Diagonal_X_offset, -$i); } sub i_func_Depth_start { my ($self, $i) = @_; ### i_func_Depth_start(): "i=$i" return $self->{'planepath_object'}->tree_depth_to_n($i); } sub i_func_Depth_end { my ($self, $i) = @_; return $self->{'planepath_object'}->tree_depth_to_n_end($i); } #------------------------------------------------------------------------------ sub pred { my ($self, $value) = @_; ### PlanePathN pred(): $value my $planepath_object = $self->{'planepath_object'}; unless ($value == int($value)) { return 0; } my ($x,$y) = $planepath_object->n_to_xy($value) or return 0; return &{$self->{'pred_func'}} ($self, $x,$y, $value); } sub pred_func_X_axis { my ($self, $x,$y) = @_; return ($x >= $self->{'i_start'} && $y == $self->{'planepath_object'}->_NumSeq_X_axis_at_Y); } sub pred_func_Y_axis { my ($self, $x,$y) = @_; return ($x == $self->{'planepath_object'}->_NumSeq_Y_axis_at_X && $y >= $self->{'i_start'}); } sub pred_func_X_neg { my ($self, $x,$y) = @_; return ($x <= - $self->{'i_start'} && $y == 0); } sub pred_func_Y_neg { my ($self, $x,$y) = @_; return ($x == 0 && $y <= - $self->{'i_start'}); } sub pred_func_Diagonal { my ($self, $x,$y) = @_; $x -= $self->{'planepath_object'}->_NumSeq_Diagonal_X_offset; return ($x >= $self->{'i_start'} && $x == $y); } sub pred_func_Diagonal_NW { my ($self, $x,$y) = @_; return ($x <= - $self->{'i_start'} && $x == -$y); } sub pred_func_Diagonal_SW { my ($self, $x,$y) = @_; return ($x <= - $self->{'i_start'} && $x == $y); } sub pred_func_Diagonal_SE { my ($self, $x,$y) = @_; return ($x >= $self->{'i_start'} && $x == -$y); } sub pred_func_Depth_start { my ($self, $x,$y, $n) = @_; return path_tree_n_is_depth_start($self->{'planepath_object'}, $n); } sub pred_func_Depth_end { my ($self, $x,$y, $n) = @_; return path_tree_n_is_depth_end($self->{'planepath_object'}, $n); } # Return true if $n is the start of a depth level. sub path_tree_n_is_depth_start { my ($path, $n) = @_; my $depth = $path->tree_n_to_depth($n); return (defined $depth && $n == $path->tree_depth_to_n($depth)); } # Return true if $n is the end of a depth level. sub path_tree_n_is_depth_end { my ($path, $n) = @_; my $depth = $path->tree_n_to_depth($n); return (defined $depth && $n == $path->tree_depth_to_n_end($depth)); } #------------------------------------------------------------------------------ use constant characteristic_integer => 1; # integer Ns sub characteristic_increasing { my ($self) = @_; ### PlanePathN characteristic_increasing(): $self my $method = "_NumSeq_$self->{'line_type'}_increasing"; my $planepath_object = $self->{'planepath_object'}; ### planepath_object: ref $planepath_object ### $method ### can code: $planepath_object->can($method) ### result: $planepath_object->can($method) && $planepath_object->$method() return $planepath_object->can($method) && $planepath_object->$method(); } sub characteristic_increasing_from_i { my ($self) = @_; ### PlanePathN characteristic_increasing_from_i(): $self my $planepath_object = $self->{'planepath_object'}; my $method = "_NumSeq_$self->{'line_type'}_increasing_from_i"; ### $method if ($method = $planepath_object->can($method)) { ### can: $method return $planepath_object->$method(); } return ($self->characteristic('increasing') ? $self->i_start : undef); } sub characteristic_non_decreasing { my ($self) = @_; ### PlanePathN characteristic_non_decreasing() ... my $planepath_object = $self->{'planepath_object'}; my $method = "_NumSeq_$self->{'line_type'}_non_decreasing"; return (($planepath_object->can($method) && $planepath_object->$method()) || $self->characteristic_increasing); } sub values_min { my ($self) = @_; ### PlanePathN values_min() ... my $method = "_NumSeq_$self->{'line_type'}_min"; my $planepath_object = $self->{'planepath_object'}; if (my $coderef = $planepath_object->can($method)) { ### $coderef return $planepath_object->$coderef(); } return $self->ith($self->i_start); } sub values_max { my ($self) = @_; my $method = "_NumSeq_$self->{'line_type'}_max"; my $planepath_object = $self->{'planepath_object'}; if (my $coderef = $planepath_object->can($method)) { return $planepath_object->$coderef(); } return undef; } { package Math::PlanePath; sub _NumSeq_X_axis_i_start { my ($self) = @_; my $x_minimum = $self->x_minimum; if (defined $x_minimum && $x_minimum > 0) { return int($x_minimum); } return 0; } sub _NumSeq_Y_axis_i_start { my ($self) = @_; ### _NumSeq_Y_axis_i_start() ... my $y_minimum = $self->y_minimum; if (defined $y_minimum && $y_minimum > 0) { return int($y_minimum); } return 0; } sub _NumSeq_Y_axis_at_X { my ($self) = @_; my $x_minimum = $self->x_minimum; if (defined $x_minimum && $x_minimum > 0) { return $x_minimum; } return 0; } sub _NumSeq_X_axis_at_Y { my ($self) = @_; my $y_minimum = $self->y_minimum; if (defined $y_minimum && $y_minimum > 0) { return $y_minimum; } return 0; } # i_start = Xminimum - Xoffset # gives Xstart = i_start + Xoffset = Xminimum to start at Xminimum use List::Util; sub _NumSeq_Diagonal_i_start { my ($self) = @_; my $x_minimum = $self->x_minimum; my $y_minimum = $self->y_minimum; return List::Util::max (int(($x_minimum||0) - $self->_NumSeq_Diagonal_X_offset), int($y_minimum||0), 0); } use constant _NumSeq_Diagonal_X_offset => 0; use constant _NumSeq_N_oeis_anum => {}; use constant _NumSeq_N_oeis_all_anum => {}; use constant _NumSeq_Depth_start_increasing => 1; use constant _NumSeq_Depth_end_increasing => 1; # sub _NumSeq_pred_X_axis { # my ($path, $value) = @_; # return ($value == int($value) # && ($path->x_negative || $value >= 0)); # } # sub _NumSeq_pred_Y_axis { # my ($path, $value) = @_; # return ($value == int($value) # && ($path->y_negative || $value >= 0)); # } } { package Math::PlanePath::SquareSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; sub _NumSeq_X_neg_increasing { my ($self) = @_; return ($self->{'wider'} == 0); } sub _NumSeq_X_neg_increasing_from_i { my ($self) = @_; ### SquareSpiral _NumSeq_X_neg_increasing_from_i(): $self # wider=0 from X=0 # wider=1 from X=-1 # wider=2 from X=-1 return int(($self->{'wider'}+1)/2); } use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; sub _NumSeq_X_neg_min { # not the value at X=0,Y=0 if wider>0 my ($self) = @_; return $self->n_start; } use constant _NumSeq_N_oeis_anum => { 'wider=0,n_start=1' => { X_axis => 'A054552', # spoke E, 4n^2 - 3n + 1 Y_neg => 'A033951', # spoke S, 4n^2 + 3n + 1 Diagonal_NW => 'A053755', # 4n^2 + 1 Diagonal_SE => 'A016754', # (2n+1)^2 # OEIS-Catalogue: A054552 planepath=SquareSpiral # OEIS-Catalogue: A033951 planepath=SquareSpiral line_type=Y_neg # OEIS-Catalogue: A053755 planepath=SquareSpiral line_type=Diagonal_NW # OEIS-Catalogue: A016754 planepath=SquareSpiral line_type=Diagonal_SE # # OEIS-Other: A054552 planepath=GreekKeySpiral,turns=0 # OEIS-Other: A033951 planepath=GreekKeySpiral,turns=0 line_type=Y_neg # OEIS-Other: A053755 planepath=GreekKeySpiral,turns=0 line_type=Diagonal_NW # OEIS-Other: A016754 planepath=GreekKeySpiral,turns=0 line_type=Diagonal_SE # Not quite, these have OFFSET=1 whereas path start X=0 # # Y_axis => 'A054556', # spoke N # # X_neg => 'A054567', # spoke W # # Diagonal => 'A054554', # spoke NE # # Diagonal_SW => 'A054569', # spoke NE # # # OEIS-Catalogue: A054556 planepath=SquareSpiral line_type=Y_axis # # # OEIS-Catalogue: A054554 planepath=SquareSpiral line_type=Diagonal }, 'wider=0,n_start=0' => { X_axis => 'A001107', Y_axis => 'A033991', Y_neg => 'A033954', # second 10-gonals Diagonal => 'A002939', Diagonal_NW => 'A016742', # 10-gonals average, 4*n^2 Diagonal_SW => 'A002943', # OEIS-Other: A001107 planepath=SquareSpiral,n_start=0 # OEIS-Catalogue: A033991 planepath=SquareSpiral,n_start=0 line_type=Y_axis # OEIS-Other: A033954 planepath=SquareSpiral,n_start=0 line_type=Y_neg # OEIS-Catalogue: A002939 planepath=SquareSpiral,n_start=0 line_type=Diagonal # OEIS-Other: A016742 planepath=SquareSpiral,n_start=0 line_type=Diagonal_NW # OEIS-Catalogue: A002943 planepath=SquareSpiral,n_start=0 line_type=Diagonal_SW }, 'wider=1,n_start=1' => { Diagonal_SW => 'A069894', # OEIS-Catalogue: A069894 planepath=SquareSpiral,wider=1 line_type=Diagonal_SW }, 'wider=1,n_start=0' => { Diagonal_SW => 'A016754', # odd squares # OEIS-Other: A016754 planepath=SquareSpiral,wider=1,n_start=0 line_type=Diagonal_SW }, }; } { package Math::PlanePath::GreekKeySpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; sub _NumSeq_X_neg_increasing { my ($self) = @_; return ($self->{'turns'} == 0); # when SquareSpiral style } *_NumSeq_Y_neg_increasing = \&_NumSeq_X_neg_increasing; sub _NumSeq_Diagonal_increasing { my ($self) = @_; return ($self->{'turns'} <= 1); } sub _NumSeq_Diagonal_NW_increasing { my ($self) = @_; return ($self->{'turns'} == 0); } *_NumSeq_Diagonal_SW_increasing = \&_NumSeq_Diagonal_increasing; sub _NumSeq_Diagonal_SE_increasing { my ($self) = @_; return ($self->{'turns'} <= 2); } use constant _NumSeq_N_oeis_anum => { 'turns=0' => (Math::PlanePath::SquareSpiral ->_NumSeq_N_oeis_anum->{'wider=0,n_start=1'} || die "Oops, SquareSpiral NumSeq PlanePathN not found"), }; } { package Math::PlanePath::PyramidSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_axis => 'A054552', # square spiral spoke E, 4n^2 - 3n + 1 Diagonal_SE => 'A033951', # square spiral spoke S, 4n^2 + 3n + 1 # OEIS-Other: A054552 planepath=PyramidSpiral # OEIS-Other: A033951 planepath=PyramidSpiral line_type=Diagonal_SE }, 'n_start=0' => { X_axis => 'A001107', # decagonal Y_axis => 'A002939', X_neg => 'A033991', Y_neg => 'A002943', Diagonal_SW => 'A007742', Diagonal_SE => 'A033954', # decagonal second kind # OEIS-Other: A001107 planepath=PyramidSpiral,n_start=0 # OEIS-Other: A002939 planepath=PyramidSpiral,n_start=0 line_type=Y_axis # OEIS-Other: A033991 planepath=PyramidSpiral,n_start=0 line_type=X_neg # OEIS-Other: A002943 planepath=PyramidSpiral,n_start=0 line_type=Y_neg # OEIS-Other: A007742 planepath=PyramidSpiral,n_start=0 line_type=Diagonal_SW # OEIS-Other: A033954 planepath=PyramidSpiral,n_start=0 line_type=Diagonal_SE }, 'n_start=2' => { Diagonal_SE => 'A185669', # OEIS-Catalogue: A185669 planepath=PyramidSpiral,n_start=2 line_type=Diagonal_SE }, }; } { package Math::PlanePath::TriangleSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_axis => 'A117625', # step by 2 each time Y_neg => 'A006137', # step by 2 each time Diagonal_SW => 'A064225', Diagonal_SE => 'A081267', # OEIS-Other: A117625 planepath=TriangleSpiral # OEIS-Other: A064225 planepath=TriangleSpiral line_type=Diagonal_SW # OEIS-Other: A081267 planepath=TriangleSpiral line_type=Diagonal_SE # # Not quite, starts value=3 at n=0 which is path Y=1 # Diagonal => 'A064226', # and duplicate in A081269 }, 'n_start=0' => { Y_axis => 'A062741', # 3*pentagonal, Y even Diagonal => 'A062708', # reading in direction 0,2,... Diagonal_SW => 'A062725', # reading in direction 0,7,... Diagonal_SE => 'A062728', # 11-gonal "second" per Math::NumSeq::Polygonal # OEIS-Catalogue: A062741 planepath=TriangleSpiral,n_start=0 line_type=Y_axis # OEIS-Catalogue: A062708 planepath=TriangleSpiral,n_start=0 line_type=Diagonal # OEIS-Catalogue: A062725 planepath=TriangleSpiral,n_start=0 line_type=Diagonal_SW # OEIS-Other: A062728 planepath=TriangleSpiral,n_start=0 line_type=Diagonal_SE # but spaced 2 apart ... # X_axis => 'A051682', # 11-gonals per Math::NumSeq::Polygonal # # OEIS-Other: A051682 planepath=TriangleSpiral,n_start=0 # X_axis }, }; } { package Math::PlanePath::TriangleSpiralSkewed; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; # ENHANCE-ME: All these variously rotated for skew=right,up,down use constant _NumSeq_N_oeis_anum => { 'skew=left,n_start=1' => { X_axis => 'A117625', X_neg => 'A006137', Y_neg => 'A064225', Diagonal => 'A081589', Diagonal_SW => 'A038764', Diagonal_SE => 'A081267', # OEIS-Catalogue: A117625 planepath=TriangleSpiralSkewed # OEIS-Catalogue: A006137 planepath=TriangleSpiralSkewed line_type=X_neg # OEIS-Catalogue: A064225 planepath=TriangleSpiralSkewed line_type=Y_neg # OEIS-Catalogue: A081589 planepath=TriangleSpiralSkewed line_type=Diagonal # OEIS-Catalogue: A038764 planepath=TriangleSpiralSkewed line_type=Diagonal_SW # OEIS-Catalogue: A081267 planepath=TriangleSpiralSkewed line_type=Diagonal_SE # OEIS-Catalogue: A081274 planepath=TriangleSpiralSkewed line_type=Diagonal_SW # # Not quite, starts OFFSET=0 value=3 but that is at path Y=1 # Y_axis => 'A064226', # and duplicate in A081269 }, 'skew=left,n_start=0' => { X_axis => 'A051682', # 11-gonals per Math::NumSeq::Polygonal Y_axis => 'A062708', # reading in direction 0,2,... Y_neg => 'A062725', # reading in direction 0,7,... Diagonal_SE => 'A062728', # 11-gonal "second" per Math::NumSeq::Polygonal Diagonal_SW => 'A081266', # OEIS-Other: A051682 planepath=TriangleSpiralSkewed,n_start=0 # X_axis # OEIS-Other: A062708 planepath=TriangleSpiralSkewed,n_start=0 line_type=Y_axis # OEIS-Other: A062725 planepath=TriangleSpiralSkewed,n_start=0 line_type=Y_neg # OEIS-Other: A062728 planepath=TriangleSpiralSkewed,n_start=0 line_type=Diagonal_SE # OEIS-Catalogue: A081266 planepath=TriangleSpiralSkewed,n_start=0 line_type=Diagonal_SW }, }; } { package Math::PlanePath::DiamondSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_axis => 'A130883', # 2*n^2-n+1 X_neg => 'A084849', # 2*n^2+n+1 Y_axis => 'A058331', # 2*n^2 + 1 Y_neg => 'A001844', # centred squares 2n(n+1)+1 # OEIS-Catalogue: A130883 planepath=DiamondSpiral # OEIS-Other: A084849 planepath=DiamondSpiral line_type=X_neg # OEIS-Catalogue: A058331 planepath=DiamondSpiral line_type=Y_axis # OEIS-Other: A001844 planepath=DiamondSpiral line_type=Y_neg }, 'n_start=0' => { X_axis => 'A000384', # 2*n^2-n, hexagonal numbers X_neg => 'A014105', # 2*n^2+n, hexagonal numbers second kind Y_axis => 'A001105', # 2*n^2 Y_neg => 'A046092', # 2n(n+1) = 4*triangular # OEIS-Other: A000384 planepath=DiamondSpiral,n_start=0 # OEIS-Other: A014105 planepath=DiamondSpiral,n_start=0 line_type=X_neg # OEIS-Other: A001105 planepath=DiamondSpiral,n_start=0 line_type=Y_axis # OEIS-Other: A046092 planepath=DiamondSpiral,n_start=0 line_type=Y_neg }, }; } { package Math::PlanePath::DiamondArms; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::AztecDiamondRings; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_axis => 'A001844', # centred squares 2n(n+1)+1 # OEIS-Other: A001844 planepath=AztecDiamondRings # Not quite, A000384 has extra value=0 # Y_axis => 'A000384', # hexagonal numbers }, 'n_start=0' => { X_axis => 'A046092', # 4*triangular Diagonal => 'A139277', # x*(8*x+5) # OEIS-Other: A046092 planepath=AztecDiamondRings,n_start=0 # OEIS-Other: A139277 planepath=AztecDiamondRings,n_start=0 line_type=Diagonal }, }; } { package Math::PlanePath::PentSpiral; use constant _NumSeq_X_axis_step => 2; # not A2 grid, but X axis steps by 2 use constant _NumSeq_X_neg_step => 2; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_axis => 'A192136', # (5*n^2-3*n+2)/2 X_neg => 'A116668', # (5n^2 + n + 2)/2 Diagonal_SE => 'A005891', # centred pentagonal (5n^2+5n+2)/2 # OEIS-Other: A192136 planepath=PentSpiral # OEIS-Other: A116668 planepath=PentSpiral line_type=X_neg # OEIS-Other: A005891 planepath=PentSpiral line_type=Diagonal_SE # Not quite, A134238 OFFSET=1 vs start X=0 here # Diagonal_SW => 'A134238', }, 'n_start=0' => { X_axis => 'A000566', # heptagonals Y_axis => 'A005476', Diagonal_SE => 'A028895', # 5*triangular # OEIS-Other: A000566 planepath=PentSpiral,n_start=0 # OEIS-Other: A005476 planepath=PentSpiral,n_start=0 line_type=Y_axis # OEIS-Other: A028895 planepath=PentSpiral,n_start=0 line_type=Diagonal_SE }, }; } { package Math::PlanePath::PentSpiralSkewed; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_axis => 'A192136', # (5*n^2-3*n+2)/2 X_neg => 'A116668', # (5n^2 + n + 2)/2 Diagonal_NW => 'A158187', # 10*n^2 + 1 Diagonal_SE => 'A005891', # centred pentagonal (5n^2+5n+2)/2 # OEIS-Catalogue: A192136 planepath=PentSpiralSkewed # OEIS-Catalogue: A116668 planepath=PentSpiralSkewed line_type=X_neg # OEIS-Catalogue: A158187 planepath=PentSpiralSkewed line_type=Diagonal_NW # OEIS-Catalogue: A005891 planepath=PentSpiralSkewed line_type=Diagonal_SE # Not quite, A140066 OFFSET=1 but path start Y=0 here # Y_axis => 'A140066', # (5n^2-11n+8)/2 but from Y=0 so using (n-1) # Not quite, A134238 OFFSET=1 but path start Y=0 here # Y_neg => 'A134238', # # OEIS-Catalogue: A134238 planepath=PentSpiralSkewed line_type=Y_neg }, 'n_start=0' => { X_axis => 'A000566', # heptagonals Y_axis => 'A005476', X_neg => 'A005475', Diagonal_NW => 'A033583', # 10*n^2 Diagonal_SE => 'A028895', # 5*triangular # OEIS-Other: A000566 planepath=PentSpiralSkewed,n_start=0 # OEIS-Catalogue: A005476 planepath=PentSpiralSkewed,n_start=0 line_type=Y_axis # OEIS-Catalogue: A005475 planepath=PentSpiralSkewed,n_start=0 line_type=X_neg # OEIS-Other: A033583 planepath=PentSpiralSkewed,n_start=0 line_type=Diagonal_NW # OEIS-Catalogue: A028895 planepath=PentSpiralSkewed,n_start=0 line_type=Diagonal_SE # Not quite, A147875 OFFSET=1 vs start Y=0 here # Y_neg => 'A147875', # second heptagonals # # OEIS-Other: A147875 planepath=PentSpiralSkewed,n_start=0 line_type=Y_neg }, }; } { package Math::PlanePath::HexSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; *_NumSeq_X_neg_increasing = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing; *_NumSeq_X_neg_increasing_from_i = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing_from_i; *_NumSeq_X_neg_min = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_min; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'wider=0,n_start=1' => { X_axis => 'A056105', # first spoke 3n^2-2n+1 Diagonal => 'A056106', # second spoke 3n^2-n+1 Diagonal_NW => 'A056107', # third spoke 3n^2+1 X_neg => 'A056108', # fourth spoke 3n^2+n+1 Diagonal_SW => 'A056109', # fifth spoke 3n^2+2n+1 Diagonal_SE => 'A003215', # centred hexagonal numbers # OEIS-Other: A056105 planepath=HexSpiral # OEIS-Other: A056106 planepath=HexSpiral line_type=Diagonal # OEIS-Other: A056107 planepath=HexSpiral line_type=Diagonal_NW # OEIS-Other: A056108 planepath=HexSpiral line_type=X_neg # OEIS-Other: A056109 planepath=HexSpiral line_type=Diagonal_SW # OEIS-Other: A003215 planepath=HexSpiral line_type=Diagonal_SE }, 'wider=0,n_start=0' => { X_axis => 'A000567', # octagonal numbers X_neg => 'A049451', Diagonal => 'A049450', Diagonal_NW => 'A033428', # octagonal numbers first,second average Diagonal_SW => 'A045944', # octagonal numbers second Diagonal_SE => 'A028896', # OEIS-Other: A000567 planepath=HexSpiral,n_start=0 # OEIS-Other: A049451 planepath=HexSpiral,n_start=0 line_type=X_neg # OEIS-Catalogue: A049450 planepath=HexSpiral,n_start=0 line_type=Diagonal # OEIS-Other: A033428 planepath=HexSpiral,n_start=0 line_type=Diagonal_NW # OEIS-Other: A045944 planepath=HexSpiral,n_start=0 line_type=Diagonal_SW # OEIS-Catalogue: A028896 planepath=HexSpiral,n_start=0 line_type=Diagonal_SE }, }; } { package Math::PlanePath::HexSpiralSkewed; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; *_NumSeq_X_neg_increasing = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing; *_NumSeq_X_neg_increasing_from_i = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing_from_i; *_NumSeq_X_neg_min = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_min; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'wider=0,n_start=1' => { X_axis => 'A056105', # first spoke 3n^2-2n+1 Y_axis => 'A056106', # second spoke 3n^2-n+1 Diagonal_NW => 'A056107', # third spoke 3n^2+1 X_neg => 'A056108', # fourth spoke 3n^2+n+1 Y_neg => 'A056109', # fifth spoke 3n^2+2n+1 Diagonal_SE => 'A003215', # centred hexagonal numbers # OEIS-Catalogue: A056105 planepath=HexSpiralSkewed # OEIS-Catalogue: A056106 planepath=HexSpiralSkewed line_type=Y_axis # OEIS-Catalogue: A056108 planepath=HexSpiralSkewed line_type=X_neg # OEIS-Catalogue: A056109 planepath=HexSpiralSkewed line_type=Y_neg # OEIS-Catalogue: A056107 planepath=HexSpiralSkewed line_type=Diagonal_NW # OEIS-Other: A003215 planepath=HexSpiralSkewed line_type=Diagonal_SE }, 'wider=0,n_start=0' => { X_axis => 'A000567', # octagonal numbers Y_axis => 'A049450', X_neg => 'A049451', Y_neg => 'A045944', # octagonal numbers second Diagonal => 'A062783', Diagonal_NW => 'A033428', # octagonal numbers first,second average Diagonal_SW => 'A063436', Diagonal_SE => 'A028896', # OEIS-Other: A000567 planepath=HexSpiralSkewed,n_start=0 # OEIS-Other: A049450 planepath=HexSpiralSkewed,n_start=0 line_type=Y_axis # OEIS-Catalogue: A049451 planepath=HexSpiralSkewed,n_start=0 line_type=X_neg # OEIS-Other: A045944 planepath=HexSpiralSkewed,n_start=0 line_type=Y_neg # OEIS-Catalogue: A062783 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal # OEIS-Other: A033428 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal_NW # OEIS-Catalogue: A063436 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal_SW # OEIS-Other: A028896 planepath=HexSpiralSkewed,n_start=0 line_type=Diagonal_SE }, # wider=1 X_axis almost 3*n^2 but not initial X=0 value # wider=1 Y_axis almost A049451 twice pentagonal but not initial X=0 # wider=2 Y_axis almost A028896 6*triangular but not initial Y=0 }; } { package Math::PlanePath::HexArms; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::HeptSpiralSkewed; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { # 'n_start=1' => # { # # Not quite, OFFSET=1 vs path start X=Y=0 # # Y_axis => 'A140065', # (7n^2 - 17n + 12)/2 but starting Y=0 not n=1 # # Diagonal_NW => 'A140063', # # Diagonal_SE => 'A069099', # }, 'n_start=0' => { X_axis => 'A001106', # 9-gonals Y_axis => 'A218471', X_neg => 'A022265', Y_neg => 'A179986', # second 9-gonals Diagonal => 'A195023', Diagonal_NW => 'A022264', Diagonal_SW => 'A186029', Diagonal_SE => 'A024966', # OEIS-Other: A001106 planepath=HeptSpiralSkewed,n_start=0 # OEIS-Catalogue: A022265 planepath=HeptSpiralSkewed,n_start=0 line_type=X_neg # OEIS-Catalogue: A218471 planepath=HeptSpiralSkewed,n_start=0 line_type=Y_axis # OEIS-Other: A179986 planepath=HeptSpiralSkewed,n_start=0 line_type=Y_neg # OEIS-Catalogue: A195023 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal # OEIS-Catalogue: A022264 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal_NW # OEIS-Catalogue: A186029 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal_SW # OEIS-Catalogue: A024966 planepath=HeptSpiralSkewed,n_start=0 line_type=Diagonal_SE }, }; } { package Math::PlanePath::OctagramSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { Diagonal_SE => 'A194268', # OEIS-Other: A194268 planepath=OctagramSpiral line_type=Diagonal_SE # Not quite, but A125201 doesn't have initial N=1 for path origin # X_axis => 'A125201' }, 'n_start=0' => { X_axis => 'A051870', # 18-gonals Y_axis => 'A139273', X_neg => 'A139275', Y_neg => 'A139277', Diagonal => 'A139272', Diagonal_NW => 'A139274', Diagonal_SW => 'A139276', Diagonal_SE => 'A139278', # second 18-gonals # OEIS-Other: A051870 planepath=OctagramSpiral,n_start=0 # OEIS-Catalogue: A139273 planepath=OctagramSpiral,n_start=0 line_type=Y_axis # OEIS-Catalogue: A139275 planepath=OctagramSpiral,n_start=0 line_type=X_neg # OEIS-Catalogue: A139277 planepath=OctagramSpiral,n_start=0 line_type=Y_neg # OEIS-Catalogue: A139272 planepath=OctagramSpiral,n_start=0 line_type=Diagonal # OEIS-Catalogue: A139274 planepath=OctagramSpiral,n_start=0 line_type=Diagonal_NW # OEIS-Catalogue: A139276 planepath=OctagramSpiral,n_start=0 line_type=Diagonal_SW # OEIS-Other: A139278 planepath=OctagramSpiral,n_start=0 line_type=Diagonal_SE }, }; } { package Math::PlanePath::AnvilSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; *_NumSeq_X_neg_increasing = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing; *_NumSeq_X_neg_increasing_from_i = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_increasing_from_i; *_NumSeq_X_neg_min = \&Math::PlanePath::SquareSpiral::_NumSeq_X_neg_min; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'wider=0,n_start=1' => { X_axis => 'A033570', # odd pentagonals (2n+1)*(3n+1) Diagonal => 'A033568', # odd second pentagonals Diagonal_SE => 'A085473', # OEIS-Catalogue: A033570 planepath=AnvilSpiral # OEIS-Catalogue: A033568 planepath=AnvilSpiral line_type=Diagonal # OEIS-Catalogue: A085473 planepath=AnvilSpiral line_type=Diagonal_SE # Not quite, A136392 OFFSET=1 value=1 whereas path start Y=0 value=1 # Y_neg => 'A136392', # 1,9,29,61, 6n^2-10n+5 }, 'wider=0,n_start=1,i_start=1' => { Y_axis => 'A126587', # points within 3,4,5 triangle, starting value=3 # OEIS-Catalogue: A126587 planepath=AnvilSpiral line_type=Y_axis i_start=1 }, 'wider=0,n_start=0' => { X_axis => 'A211014', # 14-gonal second Y_axis => 'A139267', # 2*octagonal X_neg => 'A049452', # alternate pentagonals Y_neg => 'A033580', # 4*second pentagonals Diagonal => 'A051866', # 14-gonals Diagonal_NW => 'A094159', # 3*hexagonal Diagonal_SW => 'A049453', Diagonal_SE => 'A195319', # 3*second hexagonal # OEIS-Other: A211014 planepath=AnvilSpiral,n_start=0 # OEIS-Other: A051866 planepath=AnvilSpiral,n_start=0 line_type=Diagonal # OEIS-Catalogue: A139267 planepath=AnvilSpiral,n_start=0 line_type=Y_axis # OEIS-Catalogue: A049452 planepath=AnvilSpiral,n_start=0 line_type=X_neg # OEIS-Catalogue: A033580 planepath=AnvilSpiral,n_start=0 line_type=Y_neg # OEIS-Catalogue: A094159 planepath=AnvilSpiral,n_start=0 line_type=Diagonal_NW # OEIS-Catalogue: A049453 planepath=AnvilSpiral,n_start=0 line_type=Diagonal_SW # OEIS-Catalogue: A195319 planepath=AnvilSpiral,n_start=0 line_type=Diagonal_SE }, # Not quite, A051866 starts value=0 which is at X=-1 # 'wider=1,n_start=0' => # { X_axis => 'A051866', # # OEIS-Other: A051866 planepath=AnvilSpiral,wider=1,n_start=0 # }, # 'wider=1,n_start=0' => # { Diagonal_NW_minus_1 => 'A033569', # (2*n-1)*(3*n+1) starting -1 # # XX-Other: A033569 planepath=AnvilSpiral,wider=1,n_start=-1 line_type=Diagonal_NW # }, # 'wider=2,n_start=1' => # { # Not quite, A033581 initial value=2 whereas path start N=0 # # Y_axis => 'A033581', # 6*n^2 is 14-gonals pairs average in Math::NumSeq::Polygonal # # # OEIS-Other: A033581 planepath=AnvilSpiral,wider=2 line_type=Y_axis # }, }; } { package Math::PlanePath::KnightSpiral; use constant _NumSeq_Diagonal_increasing => 1; # low then high use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::CretanLabyrinth; use constant _NumSeq_X_axis_increasing => 1; } { package Math::PlanePath::SquareArms; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::SacksSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; # SacksSpiral X_axis -- squares (i-1)^2, starting from i=1 value=0 } { package Math::PlanePath::VogelFloret; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::TheodorusSpiral; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::ArchimedeanChords; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::MultipleRings; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::PixelRings; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # where covered use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::FilledRings; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; # use constant _NumSeq_N_oeis_anum => # { # # Not quite, A036704 OFFSET=0 value=1,9,21 vs X=0 value=0,1,9,21 # 'n_start=0' => # { X_axis => 'A036704', # count points norm <= n+1/2 # }, # }; } { package Math::PlanePath::Hypot; sub _NumSeq_X_axis_i_start { my ($self) = @_; ### _NumSeq_X_axis_i_start() ... return ($self->{'points'} eq 'odd' ? 1 # X=0,Y=0 not visited : 0); } sub _NumSeq_X_neg_i_start { my ($self) = @_; ### _NumSeq_X_axis_i_start() ... return ($self->{'points'} eq 'odd' ? -1 # X=0,Y=0 not visited : 0); } *_NumSeq_Y_axis_i_start = \&_NumSeq_X_axis_i_start; *_NumSeq_Y_neg_i_start = \&_NumSeq_X_neg_i_start; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'points=all,n_start=0' => { X_axis => 'A051132', # count points < n^2 # OEIS-Catalogue: A051132 planepath=Hypot,n_start=0 }, }; } { package Math::PlanePath::HypotOctant; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; *_NumSeq_X_axis_i_start = \&Math::PlanePath::Hypot::_NumSeq_X_axis_i_start; use constant _NumSeq_N_oeis_anum => { 'points=even' => { Diagonal => 'A036702', # count points |z|<=n for 0<=b<=a # OEIS-Catalogue: A036702 planepath=HypotOctant,points=even line_type=Diagonal }, }; } { package Math::PlanePath::TriangularHypot; sub _NumSeq_X_axis_i_start { my ($self) = @_; return ($self->{'points'} eq 'odd' || $self->{'points'} eq 'hex_centred' ? 1 # X=0,Y=0 not visited : 0); } *_NumSeq_Diagonal_i_start = \&_NumSeq_X_axis_i_start; *_NumSeq_Diagonal_SE_i_start = \&_NumSeq_X_axis_i_start; sub _NumSeq_X_neg_i_start { my ($self) = @_; return - $self->_NumSeq_X_axis_i_start; } *_NumSeq_Diagonal_NW_i_start = \&_NumSeq_X_axis_i_start; *_NumSeq_Diagonal_SW_i_start = \&_NumSeq_X_axis_i_start; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::PythagoreanTree; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_N_oeis_all_anum => { Depth_start => 'A007051', # (3^n+1)/2 # OEIS-Catalogue: A007051 planepath=PythagoreanTree line_type=Depth_start # Not quite, Depth_end=(3^(n+1)-1)/2, so is n+1 # Depth_end => 'A003462', # (3^n-1)/2 }; } { package Math::PlanePath::RationalsTree; use constant _NumSeq_X_axis_increasing => 1; sub _NumSeq_Y_axis_increasing { my ($self) = @_; return ($self->{'tree_type'} eq 'L' ? 0 : 1); } sub _NumSeq_Y_axis_increasing_from_i { my ($self) = @_; return ($self->{'tree_type'} eq 'L' ? 2 : 1); } use constant _NumSeq_Y_axis_min => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'tree_type=SB' => { Depth_start => 'A000079', # powers-of-2 # RationalsTree SB -- X_axis 2^n-1 but starting X=1 # RationalsTree SB,CW -- Y_axis A000079 2^n but starting Y=1 }, 'tree_type=CW' => { Depth_start => 'A000079', # powers-of-2 }, 'tree_type=Bird' => { X_axis => 'A081254', # local max sumdisttopow2(m)/m^2 Depth_start => 'A000079', # powers-of-2 # OEIS-Catalogue: A081254 planepath=RationalsTree,tree_type=Bird # OEIS-Other: A000079 planepath=RationalsTree,tree_type=Bird line_type=Depth_start # RationalsTree Bird -- Y_axis almost A000975 10101 101010 no # consecutive equal bits, but start=1 }, 'tree_type=Drib' => { X_axis => 'A086893', # pos of fibonacci F(n+1)/F(n) in Stern diatomic Depth_start => 'A000079', # powers-of-2 # OEIS-Catalogue: A086893 planepath=RationalsTree,tree_type=Drib # Drib Y_axis # Not quite, A061547 OFFSET=1 value=0 cf path Y=1 value N=1 # Y_axis => 'A061547'# derangements or alternating bits plus pow4 }, 'tree_type=AYT' => { Depth_start => 'A000079', # powers-of-2 # RationalsTree AYT -- Y_axis A083318 2^n+1 but starting Y=1 }, 'tree_type=HCS' => { Depth_start => 'A000079', # powers-of-2 # RationalsTree HCS # Not quite, A000079 OFFSET=0 value=1 cf here X=1 N=1 # X_axis => 'A000079', # powers 2^X # Not quite, A007283 OFFSET=0 and doesn't have extra N=1 at Y=1 # Y_axis => 'A007283', # 3*2^n starting OFFSET=0 value=3 }, }; } { package Math::PlanePath::FractionsTree; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_X_offset => -1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Y_axis_i_start => 2; } { package Math::PlanePath::ChanTree; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_X_axis_i_start => 1; use constant _NumSeq_Y_axis_i_start => 1; # start at Y=1 use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'k=2,n_start=1' => { Depth_start => 'A000079', # powers-of-2 # OEIS-Other: A000079 planepath=ChanTree,n_start=1,k=2 line_type=Depth_start # Depth_end is 2^k-1 A000225, or 2^k-2 A000918, but without initial # 0 or -1. }, 'k=2,n_start=0' => { Depth_start => 'A000225', # 2^k-1 # OEIS-Other: A000225 planepath=ChanTree,k=2 line_type=Depth_start }, 'k=3,n_start=1' => { Depth_start => 'A000244', # powers-of-3 # OEIS-Other: A000244 planepath=ChanTree,n_start=1 line_type=Depth_start }, 'k=4,n_start=1' => { Depth_start => 'A000302', # powers-of-4 # OEIS-Other: A000302 planepath=ChanTree,n_start=1,k=4 line_type=Depth_start }, 'k=5,n_start=1' => { Depth_start => 'A000351', # powers-of-5 # OEIS-Other: A000351 planepath=ChanTree,n_start=1,k=5 line_type=Depth_start }, 'k=10,n_start=1' => { Depth_start => 'A011557', # powers-of-10 # OEIS-Other: A011557 planepath=ChanTree,n_start=1,k=10 line_type=Depth_start }, }; } { package Math::PlanePath::DiagonalRationals; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_axis_i_start => 1; use constant _NumSeq_Y_axis_i_start => 1; # Diagonal => 'A002088', # cumulative totient # Not quite, start X=1 value=1 cf seq OFFSET=0 value=0 } { package Math::PlanePath::FactorRationals; use constant _NumSeq_X_axis_i_start => 1; sub _NumSeq_X_axis_increasing { my ($self) = @_; # perfect squares along X axis of even/odd return $self->{'factor_coding'} eq 'even/odd'; } use constant _NumSeq_Y_axis_i_start => 1; sub _NumSeq_Y_axis_increasing { my ($self) = @_; # perfect squares along Y axis of odd/even return $self->{'factor_coding'} eq 'odd/even'; } use constant _NumSeq_N_oeis_anum => { 'factor_coding=even/odd' => { Y_axis => 'A102631', # n^2/(squarefree kernel) # OEIS-Catalogue: A102631 planepath=FactorRationals line_type=Y_axis # # Not quite, OFFSET=0 value 0 whereas start X=Y=1 value 1 here # X_axis => 'A000290', # squares 0,1,4,9 # # OEIS-Other: A000290 planepath=FactorRationals }, 'factor_coding=odd/even' => { X_axis => 'A102631', # n^2/(squarefree kernel) # OEIS-Other: A102631 planepath=FactorRationals,factor_coding=odd/even }, }; } { package Math::PlanePath::GcdRationals; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_axis_i_start => 1; use constant _NumSeq_Y_axis_i_start => 1; # GcdRationals # Not quite, starts X=1 # X_axis => triangular row # Not quite, starts X=1 here cf OFFSET=0 in A000124 # Y_axis => 'A000124', # triangular+1 # # GcdRationals,pairs_order=diagonals_down # Not quite, start X=1 here cf A000290 starts OFFSET=1 # X_axis => 'A000290', # Y=1 row, perfect squares # Not quite, A033638 starts two ones 1,1,... # Y_axis => 'A033638', # quarter-squares + 1 # # GcdRationals,pairs_order=diagonals_up # Not quite, A002061 starts two ones 1,1, # X_axis => 'A002061', # Not quite, X=1 column squares+pronic, but no initial 0,0 of A002620 # Y_axis => 'A002620', # X=1 column # Not quite, starting value=2 here # Diagonal_above => 'A002522', # Y=X+1 diagonal, squares+1 } { package Math::PlanePath::CfracDigits; # 1 # diagonal Y/(Y+1) = 0 + ----- # 1 + 1/Y # q0=1 q1=Y # N = 3,Y-1 in 1,2,3 # = 1,0,Y-1 in 0,1,2 # use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_X_offset => -1; use constant _NumSeq_Y_axis_increasing => 1; # radix without digit 0 } { package Math::PlanePath::PeanoCurve; sub _NumSeq_X_axis_increasing { my ($self) = @_; return ($self->{'radix'} % 2); } *_NumSeq_Y_axis_increasing = \&_NumSeq_X_axis_increasing; use constant _NumSeq_N_oeis_anum => { 'radix=3' => { X_axis => 'A163480', # axis same as initial direction Y_axis => 'A163481', # axis opp to initial direction Diagonal => 'A163343', }, # OEIS-Catalogue: A163480 planepath=PeanoCurve # OEIS-Catalogue: A163481 planepath=PeanoCurve line_type=Y_axis # OEIS-Catalogue: A163343 planepath=PeanoCurve line_type=Diagonal # OEIS-Other: A163480 planepath=GrayCode,apply_type=TsF,radix=3 # OEIS-Other: A163481 planepath=GrayCode,apply_type=TsF,radix=3 line_type=Y_axis # OEIS-Other: A163343 planepath=GrayCode,apply_type=TsF,radix=3 line_type=Diagonal # OEIS-Other: A163480 planepath=GrayCode,apply_type=FsT,radix=3 # OEIS-Other: A163481 planepath=GrayCode,apply_type=FsT,radix=3 line_type=Y_axis # OEIS-Other: A163343 planepath=GrayCode,apply_type=FsT,radix=3 line_type=Diagonal }; } # { package Math::PlanePath::PeanoDiagonals; # } { package Math::PlanePath::WunderlichSerpentine; sub _NumSeq_X_axis_increasing { my ($self) = @_; if ($self->{'radix'} % 2) { return 1; # odd radix always increasing } # FIXME: depends on the serpentine_type bits return 0; } sub _NumSeq_Y_axis_increasing { my ($self) = @_; if ($self->{'radix'} % 2) { return 1; # odd radix always increasing } # FIXME: depends on the serpentine_type bits return 0; } } { package Math::PlanePath::HilbertCurve; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { '' => { X_axis => 'A163482', Y_axis => 'A163483', Diagonal => 'A062880', # base 4 digits 0,2 only # OEIS-Catalogue: A163482 planepath=HilbertCurve # OEIS-Catalogue: A163483 planepath=HilbertCurve line_type=Y_axis # OEIS-Other: A062880 planepath=HilbertCurve line_type=Diagonal }, }; } { package Math::PlanePath::HilbertSides; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { '' => { Diagonal => 'A062880', # base 4 digits 0,2 only # OEIS-Other: A062880 planepath=HilbertSides line_type=Diagonal }, }; } { package Math::PlanePath::HilbertSpiral; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; } { package Math::PlanePath::ZOrderCurve; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'radix=2' => { X_axis => 'A000695', # base 4 digits 0,1 only (RadixConversion) Y_axis => 'A062880', # base 4 digits 0,2 only Diagonal => 'A001196', # base 4 digits 0,3 only # OEIS-Other: A000695 planepath=ZOrderCurve # OEIS-Catalogue: A062880 planepath=ZOrderCurve line_type=Y_axis # OEIS-Catalogue: A001196 planepath=ZOrderCurve line_type=Diagonal }, 'radix=3' => { X_axis => 'A037314', # base 9 digits 0,1,2 only Diagonal => 'A338086', # base 9 digits 0,4,8 only # OEIS-Catalogue: A037314 planepath=ZOrderCurve,radix=3 # OEIS-Catalogue: A338086 planepath=ZOrderCurve,radix=3 line_type=Diagonal }, 'radix=3,i_start=1' => { Y_axis => 'A208665', # base 9 digits 0,3,6 only, starting OFFSET=1 value=3 # OEIS-Catalogue: A208665 planepath=ZOrderCurve,radix=3 i_start=1 line_type=Y_axis # ZOrderCurve dir=2 radix,3: match 6,27,30,33,54,57,60,243,246,249,270,273,276,297,300 # A208665 Numbers that match odd ternary polynomials; see Comments. # A208665 ,3,6,27,30,33,54,57,60,243,246,249,270,273,276,297,300,303,486,489,492,513,516,519,540,543,546,2187,2190,2193,2214,2217,2220,2241,2244,2247,2430,2433,2436,2457,2460,2463,2484,2487,2490,2673,2676,2679, # base 9 digits 0,3,6 only }, 'radix=10' => { X_axis => 'A051022', # base 10 insert 0s, for digits 0 to 9 base 100 Diagonal => 'A338754', # base 10 duplicate digits # OEIS-Catalogue: A051022 planepath=ZOrderCurve,radix=10 # OEIS-Catalogue: A338754 planepath=ZOrderCurve,radix=10 line_type=Diagonal }, }; } { package Math::PlanePath::GrayCode; # X axis increasing for: # radix=2 TsF,Fs # radix=3 reflected TsF,FsT # radix=3 modular TsF,Fs # radix=4 reflected TsF,Fs # radix=4 modular TsF,Fs # radix=5 reflected TsF,FsT # radix=5 modular TsF,Fs # sub _NumSeq_X_axis_increasing { my ($self) = @_; if ($self->{'gray_type'} eq 'modular' || $self->{'radix'} == 2) { return ($self->{'apply_type'} eq 'TsF' || $self->{'apply_type'} eq 'Fs'); } if ($self->{'radix'} & 1) { return ($self->{'apply_type'} eq 'TsF' || $self->{'apply_type'} eq 'FsT'); } else { return ($self->{'apply_type'} eq 'TsF' || $self->{'apply_type'} eq 'Fs'); } } *_NumSeq_Y_axis_increasing = \&_NumSeq_X_axis_increasing; # Diagonal increasing for: # radix=2 FsT,Ts # radix=3 reflected Ts,Fs # radix=3 modular FsT # radix=4 reflected FsT,Ts # radix=4 modular FsT # radix=5 reflected Ts,Fs # radix=5 modular FsT sub _NumSeq_Diagonal_increasing { my ($self) = @_; if ($self->{'radix'} & 1) { if ($self->{'gray_type'} eq 'modular') { return ($self->{'apply_type'} eq 'FsT'); # odd modular } else { return ($self->{'apply_type'} eq 'Ts' || $self->{'apply_type'} eq 'Fs'); # odd reflected } } if ($self->{'gray_type'} eq 'reflected' || $self->{'radix'} == 2) { return ($self->{'apply_type'} eq 'FsT' || $self->{'apply_type'} eq 'Ts'); # even reflected } else { return ($self->{'apply_type'} eq 'FsT'); # even modular } } use constant _NumSeq_N_oeis_anum => { 'apply_type=TsF,gray_type=reflected,radix=3' => (Math::PlanePath::PeanoCurve->_NumSeq_N_oeis_anum->{'radix=3'} || die "Oops, SquareSpiral NumSeq PlanePathN not found"), 'apply_type=FsT,gray_type=reflected,radix=3' => (Math::PlanePath::PeanoCurve->_NumSeq_N_oeis_anum->{'radix=3'} || die "Oops, SquareSpiral NumSeq PlanePathN not found"), # GrayCode radix=2 TsF==Fs reflected==modular do { my $href = { Y_axis => 'A001196', # base 4 digits 0,3 only }; ('apply_type=TsF,gray_type=reflected,radix=2' => $href, 'apply_type=Fs,gray_type=reflected,radix=2' => $href, 'apply_type=TsF,gray_type=modular,radix=2' => $href, 'apply_type=Fs,gray_type=modular,radix=2' => $href, ); # OEIS-Other: A001196 planepath=GrayCode,apply_type=TsF line_type=Y_axis # OEIS-Other: A001196 planepath=GrayCode,apply_type=Fs line_type=Y_axis # OEIS-Other: A001196 planepath=GrayCode,apply_type=TsF,gray_type=modular line_type=Y_axis # OEIS-Other: A001196 planepath=GrayCode,apply_type=Fs,gray_type=modular line_type=Y_axis }, # GrayCode radix=2 Ts==FsT reflected==modular do { my $href = { Diagonal => 'A062880', # base 4 digits 0,2 only }; ('apply_type=Ts,gray_type=reflected,radix=2' => $href, 'apply_type=Ts,gray_type=modular,radix=2' => $href, 'apply_type=FsT,gray_type=reflected,radix=2' => $href, 'apply_type=FsT,gray_type=modular,radix=2' => $href, ); # OEIS-Other: A062880 planepath=GrayCode,apply_type=Ts line_type=Diagonal # OEIS-Other: A062880 planepath=GrayCode,apply_type=Ts,gray_type=modular line_type=Diagonal # OEIS-Other: A062880 planepath=GrayCode,apply_type=FsT line_type=Diagonal # OEIS-Other: A062880 planepath=GrayCode,apply_type=FsT,gray_type=modular line_type=Diagonal }, # GrayCode radix=3 sT==sF reflected # N split then toGray giving Y=0 means N ternary 010202 etc # N split then toGray giving X=Y means N ternary pairs 112200 do { my $href = { X_axis => 'A163344', # central Peano/4, base9 digits 0,1,2 only Diagonal => 'A163343', # central diagonal of Peano, base9 0,4,8 }; ('apply_type=sT,gray_type=reflected,radix=3' => $href, 'apply_type=sF,gray_type=reflected,radix=3' => $href, ); # OEIS-Catalogue: A163344 planepath=GrayCode,apply_type=sT,radix=3 # OEIS-Other: A163344 planepath=GrayCode,apply_type=sF,radix=3 # OEIS-Other: A163343 planepath=GrayCode,apply_type=sT,radix=3 line_type=Diagonal # OEIS-Other: A163343 planepath=GrayCode,apply_type=sF,radix=3 line_type=Diagonal }, 'apply_type=FsT,gray_type=modular,radix=3,i_start=1' => { Diagonal => 'A208665', # base 9 digits 0,3,6 # OEIS-Other: A208665 planepath=GrayCode,apply_type=FsT,gray_type=modular,radix=3 line_type=Diagonal i_start=1 }, }; } # { package Math::PlanePath::ImaginaryBase; # } { package Math::PlanePath::ImaginaryHalf; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'radix=2,digit_order=YXX' => { Y_axis => 'A033045', # base 8 digits 0,1 only # OEIS-Other: A033045 planepath=ImaginaryHalf,digit_order=YXX line_type=Y_axis }, 'radix=2,digit_order=YXnX' => { Y_axis => 'A033045', # base 8 digits 0,1 only # OEIS-Other: A033045 planepath=ImaginaryHalf,digit_order=YXnX line_type=Y_axis }, }; } # { package Math::PlanePath::CubicBase; # } { package Math::PlanePath::DekkingCurve; sub _NumSeq_X_axis_increasing { my ($self) = @_; # arms=4 has various interleaving return $self->{'arms'} < 4; } sub _NumSeq_X_neg_increasing { my ($self) = @_; # arms=3 has various interleaving return $self->{'arms'} < 3; } sub _NumSeq_Y_axis_increasing { my ($self) = @_; # arms=2 has various interleaving return $self->{'arms'} < 2; } sub _NumSeq_Y_neg_increasing { my ($self) = @_; # arms=4 has various interleaving return $self->{'arms'} < 4; } } { package Math::PlanePath::DekkingCentres; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; } { package Math::PlanePath::CincoCurve; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; } { package Math::PlanePath::BetaOmega; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; } { package Math::PlanePath::KochelCurve; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; } { package Math::PlanePath::AR2W2Curve; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; } { package Math::PlanePath::WunderlichMeander; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; } # { package Math::PlanePath::Flowsnake; # } # { package Math::PlanePath::FlowsnakeCentres; # # inherit from Flowsnake # } # { package Math::PlanePath::GosperIslands; # } # { package Math::PlanePath::GosperSide; # } { package Math::PlanePath::KochCurve; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_Diagonal_increasing => 1; # when touched } { package Math::PlanePath::KochPeaks; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; # when touched # Diagonal never touched } { package Math::PlanePath::KochSnowflakes; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::KochSquareflakes; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::QuadricCurve; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Diagonal_increasing => 1; # two values only } { package Math::PlanePath::QuadricIslands; # FIXME: pred() on Diagonal_SW doesn't notice 0.5 square use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 0; use constant _NumSeq_Y_neg_increasing_from_i => 1; # after 3,2,8 use constant _NumSeq_Y_neg_min => 2; # at X=-1,Y=0 rather than X=0,Y=0 use constant _NumSeq_Diagonal_SW_increasing => 0; use constant _NumSeq_Diagonal_SW_min => 1; } { package Math::PlanePath::SierpinskiTriangle; use constant _NumSeq_X_axis_increasing => 1; # for "diagonal" style use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; # low 10111=23 increment to 11000=24 # 10111 ones=4 width=2^4 # cf A160722 is 3*A006046-2*n, drawing three Sierpinski triangles # http://www.polprimos.com/imagenespub/polca722.jpg # use constant _NumSeq_N_oeis_anum => { #--------- # i_start=0, n_start=0 'align=triangular,n_start=0' => { Diagonal_NW => 'A006046', Depth_start => 'A006046', # OEIS-Other: A006046 planepath=SierpinskiTriangle line_type=Diagonal_NW # OEIS-Other: A006046 planepath=SierpinskiTriangle line_type=Depth_start }, 'align=right,n_start=0' => { Y_axis => 'A006046', Depth_start => 'A006046', # OEIS-Catalogue: A006046 planepath=SierpinskiTriangle,align=diagonal,n_start=0 line_type=Y_axis }, 'align=left,n_start=0' => { Diagonal_NW => 'A006046', Depth_start => 'A006046', # OEIS-Other: A006046 planepath=SierpinskiTriangle,align=left,n_start=0 line_type=Diagonal_NW }, 'align=diagonal,n_start=0' => { Y_axis => 'A006046', Depth_start => 'A006046', # OEIS-Other: A006046 planepath=SierpinskiTriangle,align=diagonal,n_start=0 line_type=Y_axis }, #--------- # i_start=1, n_start=0 # starting OFFSET=1 value=2,4,8,10 so missing N=0 at Y=0, hence i_start=1 'align=triangular,n_start=0,i_start=1' => { Diagonal => 'A074330', Depth_end => 'A074330', # OEIS-Catalogue: A074330 planepath=SierpinskiTriangle line_type=Diagonal i_start=1 # OEIS-Other: A074330 planepath=SierpinskiTriangle line_type=Depth_end i_start=1 }, 'align=right,n_start=0,i_start=1' => { Diagonal => 'A074330', Depth_end => 'A074330', # OEIS-Other: A074330 planepath=SierpinskiTriangle,align=right line_type=Diagonal i_start=1 }, 'align=left,n_start=0,i_start=1' => { Y_axis => 'A074330', Depth_end => 'A074330', # OEIS-Other: A074330 planepath=SierpinskiTriangle,align=left line_type=Y_axis i_start=1 }, 'align=diagonal,n_start=0,i_start=1' => { X_axis => 'A074330', Depth_end => 'A074330', # OEIS-Other: A074330 planepath=SierpinskiTriangle,align=diagonal i_start=1 }, }; } { package Math::PlanePath::SierpinskiArrowhead; use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; # align="diagonal" is X increasing, other align is single origin point only use constant _NumSeq_X_axis_increasing => 1; } { package Math::PlanePath::SierpinskiArrowheadCentres; use constant _NumSeq_Y_axis_increasing => 1; # never touched ? use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; # align="diagonal" is X increasing, other align is single origin point only use constant _NumSeq_X_axis_increasing => 1; } { package Math::PlanePath::SierpinskiCurve; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_X_axis_i_start => 1; # but not all cells visited use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_i_start => 1; # but not all cells visited use constant _NumSeq_X_neg_increasing => 1; # arms use constant _NumSeq_Y_neg_increasing => 1; # arms use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::SierpinskiCurveStair; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_X_neg_increasing => 1; # arms use constant _NumSeq_Y_neg_increasing => 1; # arms use constant _NumSeq_Diagonal_increasing => 1; # when touched use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; } { package Math::PlanePath::HIndexing; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when touched } # { package Math::PlanePath::DragonCurve; # } # { package Math::PlanePath::DragonRounded; # } # { package Math::PlanePath::DragonMidpoint; # } { package Math::PlanePath::AlternatePaper; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; # arms use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; # selecting the smaller N on the negative axes gives increasing, maybe use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'arms=1' => { X_axis => 'A000695', # base 4 digits 0,1 only Diagonal => 'A062880', # base 4 digits 0,2 only # OEIS-Other: A000695 planepath=AlternatePaper # OEIS-Other: A062880 planepath=AlternatePaper line_type=Diagonal }, # FIXME: not sure what to do when multiple-visited points on axes # 'arms=2' => # { X_axis => 'A062880', # base 4 digits 0,2 only # Y_axis => 'A145812', # base 4 digits 0,2 only except low digit 1,3 only # # OEIS-Other: A062880 planepath=AlternatePaper,arms=2 # # OEIS-Catalogue: A145812 planepath=AlternatePaper,arms=2 line_type=Y_axis # }, # 'arms=3' => # { X_axis => 'A001196', # base 4 digits 0,3 only # # OEIS-Other: A001196 planepath=AlternatePaper,arms=3 # }, # # alt paper arms=4 diagonal, arms=8 x axis # A127988 # base 8 digits 0,4 only }; } { package Math::PlanePath::AlternatePaperMidpoint; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; # arms use constant _NumSeq_Diagonal_SE_increasing => 1; # arms } # { package Math::PlanePath::TerdragonCurve; # } # { package Math::PlanePath::TerdragonRounded; # } # { package Math::PlanePath::TerdragonMidpoint; # } # { package Math::PlanePath::AlternateTerdragon; # } # { package Math::PlanePath::R5DragonCurve; # } # { package Math::PlanePath::R5DragonMidpoint; # } # { package Math::PlanePath::CCurve; # } # { package Math::PlanePath::ComplexPlus; # } { package Math::PlanePath::ComplexMinus; use constant _NumSeq_N_oeis_anum => { 'realpart=1' => { X_axis => 'A066321', # binary base i-1 X_neg => 'A256441', # binary base i-1 # OEIS-Catalogue: A066321 planepath=ComplexMinus # OEIS-Catalogue: A256441 planepath=ComplexMinus line_type=X_neg # cf A066323 count of 1-bits in N on X axis }, }; } # { package Math::PlanePath::ComplexRevolving; # } { package Math::PlanePath::Rows; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Y_neg_min => undef; # negatives use constant _NumSeq_Y_neg_max => 1; # negatives # secret negatives # (w-1)*(w-1)-1 # = w^2-2w+1-1 # = w(w-2) sub _NumSeq_Diagonal_SE_min { my ($self) = @_; return ($self->{'width'}-2)*$self->{'width'}; } use constant _NumSeq_N_oeis_anum => { 'n_start=0,width=1' => { Y_axis => 'A001477', # integers 0,1,2,3,etc # OEIS-Other: A001477 planepath=Rows,width=1,n_start=0 line_type=Y_axis }, 'n_start=1,width=2' => { Y_axis => 'A005408', # odd 2n+1 # OEIS-Other: A005408 planepath=Rows,width=2 line_type=Y_axis }, 'n_start=1,width=3' => { Y_axis => 'A016777', # 3n+1 # OEIS-Catalogue: A016777 planepath=Rows,width=3 line_type=Y_axis }, 'n_start=1,width=4' => { Y_axis => 'A016813', # 4n+1 # OEIS-Catalogue: A016813 planepath=Rows,width=4 line_type=Y_axis }, 'n_start=1,width=5' => { Y_axis => 'A016861', # 5n+1 # OEIS-Catalogue: A016861 planepath=Rows,width=5 line_type=Y_axis }, 'n_start=1,width=6' => { Y_axis => 'A016921', # 6n+1 # OEIS-Catalogue: A016921 planepath=Rows,width=6 line_type=Y_axis }, 'n_start=1,width=7' => { Y_axis => 'A016993', # 7n+1 # OEIS-Catalogue: A016993 planepath=Rows,width=7 line_type=Y_axis }, }; } { package Math::PlanePath::Columns; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_X_neg_min => undef; # negatives use constant _NumSeq_X_neg_max => 1; # negatives sub _NumSeq_Diagonal_NW_min { my ($self) = @_; # secret negatives return ($self->{'height'}-2)*$self->{'height'}; } use constant _NumSeq_N_oeis_anum => { 'n_start=0,height=1' => { X_axis => 'A001477', # integers 0,1,2,3,etc # OEIS-Other: A001477 planepath=Columns,height=1,n_start=0 }, 'n_start=1,height=2' => { X_axis => 'A005408', # odd 2n+1 # OEIS-Other: A005408 planepath=Columns,height=2 }, 'n_start=1,height=3' => { X_axis => 'A016777', # 3n+1 # OEIS-Other: A016777 planepath=Columns,height=3 }, 'n_start=1,height=4' => { X_axis => 'A016813', # 4n+1 # OEIS-Other: A016813 planepath=Columns,height=4 }, 'n_start=1,height=5' => { X_axis => 'A016861', # 5n+1 # OEIS-Other: A016861 planepath=Columns,height=5 }, 'n_start=1,height=6' => { X_axis => 'A016921', # 6n+1 # OEIS-Other: A016921 planepath=Columns,height=6 }, 'n_start=1,height=7' => { X_axis => 'A016993', # 7n+1 # OEIS-Other: A016993 planepath=Columns,height=7 }, }; } { package Math::PlanePath::Diagonals; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'direction=down,n_start=1,x_start=0,y_start=0' => { # Diagonals X_axis -- triangular 1,3,6,etc, but starting i=0 value=1 Y_axis => 'A000124', # triangular+1 = n*(n+1)/2+1 Diagonal => 'A001844', # centred squares 2n(n+1)+1 # OEIS-Catalogue: A000124 planepath=Diagonals line_type=Y_axis # OEIS-Catalogue: A001844 planepath=Diagonals line_type=Diagonal }, 'direction=up,n_start=1,x_start=0,y_start=0' => { X_axis => 'A000124', # triangular+1 = n*(n+1)/2+1 Diagonal => 'A001844', # centred squares 2n(n+1)+1 # OEIS-Other: A000124 planepath=Diagonals,direction=up # OEIS-Other: A001844 planepath=Diagonals,direction=up line_type=Diagonal }, 'direction=down,n_start=0,x_start=0,y_start=0' => { X_axis => 'A000096', # n*(n+3)/2 Y_axis => 'A000217', # triangular n*(n+1)/2 # OEIS-Other: A000096 planepath=Diagonals,n_start=0 # OEIS-Other: A000217 planepath=Diagonals,n_start=0 line_type=Y_axis }, 'direction=up,n_start=0,x_start=0,y_start=0' => { X_axis => 'A000217', # triangular n*(n+1)/2 Y_axis => 'A000096', # n*(n+3)/2 # OEIS-Other: A000217 planepath=Diagonals,direction=up,n_start=0 # OEIS-Other: A000096 planepath=Diagonals,direction=up,n_start=0 line_type=Y_axis }, }; } { package Math::PlanePath::DiagonalsAlternating; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { Diagonal => 'A001844', # centred squares 2n(n+1)+1 # OEIS-Other: A001844 planepath=DiagonalsAlternating line_type=Diagonal # Not quite, extra initial 1 or 0 # X_axis => 'A128918', # Y_axis => 'A131179', }, 'n_start=0' => { Diagonal => 'A046092', # 2*triangular # OEIS-Other: A046092 planepath=DiagonalsAlternating,n_start=0 line_type=Diagonal }, }; } { package Math::PlanePath::DiagonalsOctant; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { # 'direction=down,n_start=1' => # { # Not quite, starting i=0 for square=1 cf A000290 starts 0 # # Diagonal => 'A000290', # squares # # Not quite, A033638 extra initial 1 # # Y_axis => 'A033638', # quarter squares + 1, 1,1,2,3,5,7,10,13 # } # 'direction=up,n_start=1' => # { # # Not quite, A002061 extra initial 1 # # Diagonal => 'A002061', # 1,1,3,7,13,21,31,43 # } 'direction=down,n_start=0' => { Diagonal => 'A005563', # n*(n+2) 0,3,8,15,24 # OEIS-Other: A005563 planepath=DiagonalsOctant,n_start=0 line_type=Diagonal # # Not quite, extra initial 0 # # Y_axis => 'A002620', # quarter squares 0,0,1,2,4,6,9,12, }, 'direction=up,n_start=0' => { Diagonal => 'A002378', # pronic 0,2,6,12,20 # OEIS-Other: A002378 planepath=DiagonalsOctant,direction=up,n_start=0 line_type=Diagonal # # Not quite, starts n=1 value=0 whereas start Y=0 value=0 here # # Y_axis => 'A024206', # 0,1,3,5,8,11,15 }, }; } { package Math::PlanePath::MPeaks; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 0; use constant _NumSeq_X_neg_increasing_from_i => 1; use constant _NumSeq_X_neg_min => 1; # at X=-1,Y=0 rather than X=0,Y=0 use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing_from_i => 1; use constant _NumSeq_Diagonal_NW_min => 2; # at X=-1,Y=1 # MPeaks -- X_axis A045944 matchstick n(3n+2) but initial N=3 # MPeaks -- Diagonal,Y_axis hexagonal first,second spoke, but starting # from 3 } { package Math::PlanePath::Staircase; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { Diagonal => 'A084849', # OEIS-Other: A084849 planepath=Staircase line_type=Diagonal }, 'n_start=0' => { Diagonal => 'A014105', # second hexagonals # OEIS-Other: A014105 planepath=Staircase,n_start=0 line_type=Diagonal }, 'n_start=2' => { Diagonal => 'A096376', # OEIS-Catalogue: A096376 planepath=Staircase,n_start=2 line_type=Diagonal # Not quite, A128918 has extra initial 1,1 # X_axis => 'A128918', }, }; } { package Math::PlanePath::StaircaseAlternating; sub _NumSeq_X_axis_increasing { my ($self) = @_; return ($self->{'end_type'} eq 'square' ? 1 : 0); # backs-up } *_NumSeq_Y_axis_increasing = \&_NumSeq_X_axis_increasing; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'end_type=jump,n_start=1' => { Diagonal => 'A084849', # OEIS-Other: A084849 planepath=StaircaseAlternating line_type=Diagonal }, 'end_type=jump,n_start=0' => { Diagonal => 'A014105', # second hexagonals # OEIS-Other: A014105 planepath=StaircaseAlternating,n_start=0 line_type=Diagonal }, 'end_type=jump,n_start=2' => { Diagonal => 'A096376', # OEIS-Other: A096376 planepath=StaircaseAlternating,n_start=2 line_type=Diagonal }, 'end_type=square,n_start=1' => { Diagonal => 'A058331', # OEIS-Other: A058331 planepath=StaircaseAlternating,end_type=square line_type=Diagonal }, 'end_type=square,n_start=0' => { Diagonal => 'A001105', # OEIS-Other: A001105 planepath=StaircaseAlternating,end_type=square,n_start=0 line_type=Diagonal }, }; } { package Math::PlanePath::Corner; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'wider=0,n_start=1' => { Y_axis => 'A002522', # n^2+1 # OEIS-Other: A002522 planepath=Corner line_type=Y_axis }, 'wider=0,n_start=0' => { X_axis => 'A005563', # (n+1)^2-1 Y_axis => 'A000290', # squares Diagonal => 'A002378', # pronic # OEIS-Other: A005563 planepath=Corner,n_start=0 # OEIS-Other: A000290 planepath=Corner,n_start=0 line_type=Y_axis # OEIS-Other: A002378 planepath=Corner,n_start=0 line_type=Diagonal }, 'wider=0,n_start=2' => { Y_axis => 'A059100', # n^2+2 Diagonal => 'A014206', # pronic+2 # OEIS-Catalogue: A059100 planepath=Corner,n_start=2 line_type=Y_axis # OEIS-Catalogue: A014206 planepath=Corner,n_start=2 line_type=Diagonal }, 'wider=1,n_start=0' => { Y_axis => 'A002378', # pronic Diagonal => 'A005563', # (n+1)^2-1 # OEIS-Other: A002378 planepath=Corner,wider=1,n_start=0 line_type=Y_axis # OEIS-Other: A005563 planepath=Corner,wider=1,n_start=0 line_type=Diagonal }, 'wider=1,n_start=2' => { Y_axis => 'A014206', # pronic+2 # OEIS-Other: A014206 planepath=Corner,wider=1,n_start=2 line_type=Y_axis }, 'wider=2,n_start=0' => { Y_axis => 'A005563', # (n+1)^2-1 Diagonal => 'A028552', # n(n+3) # OEIS-Other: A005563 planepath=Corner,wider=2,n_start=0 line_type=Y_axis # OEIS-Catalogue: A028552 planepath=Corner,wider=2,n_start=0 line_type=Diagonal }, 'wider=2,n_start=1' => { Diagonal => 'A028387', # n(n+3)+1 = n + (n+1)^2 # OEIS-Catalogue: A028387 planepath=Corner,wider=2,n_start=1 line_type=Diagonal }, 'wider=3,n_start=0' => { Y_axis => 'A028552', # n(n+3) # OEIS-Other: A028552 planepath=Corner,wider=3,n_start=0 line_type=Y_axis }, }; } { package Math::PlanePath::CornerAlternating; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'wider=0,n_start=1' => { X_axis => 'A081346', # picture in A081344 Y_axis => 'A081345', # OEIS-Catalogue: A081346 planepath=CornerAlternating # OEIS-Catalogue: A081345 planepath=CornerAlternating line_type=Y_axis }, 'wider=0,n_start=0' => { Diagonal => 'A002378', # pronic # OEIS-Other: A002378 planepath=CornerAlternating,n_start=0 line_type=Diagonal }, 'wider=0,n_start=2' => { Diagonal => 'A014206', # pronic # OEIS-Other: A014206 planepath=CornerAlternating,n_start=2 line_type=Diagonal }, 'wider=1,n_start=1' => { X_axis => 'A081347', # maze, picture in A081344 Y_axis => 'A081348', Diagonal => 'A080335', # OEIS-Catalogue: A081347 planepath=CornerAlternating,wider=1 # OEIS-Catalogue: A081348 planepath=CornerAlternating,wider=1 line_type=Y_axis # OEIS-Catalogue: A080335 planepath=CornerAlternating,wider=1 line_type=Diagonal }, 'wider=2,n_start=1' => { X_axis => 'A081350', # another maze, picture in A081349 Y_axis => 'A081351', Diagonal => 'A081352', # OEIS-Catalogue: A081350 planepath=CornerAlternating,wider=2 # OEIS-Catalogue: A081351 planepath=CornerAlternating,wider=2 line_type=Y_axis # OEIS-Catalogue: A081352 planepath=CornerAlternating,wider=2 line_type=Diagonal }, }; } { package Math::PlanePath::PyramidRows; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # when covered, or single use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_N_oeis_anum => { # PyramidRows step=1 do { my $href = { Y_axis => 'A000124', # triangular+1 = n*(n+1)/2+1 }; ('step=1,align=centre,n_start=1' => $href, 'step=1,align=right,n_start=1' => $href); # OEIS-Other: A000124 planepath=PyramidRows,step=1 line_type=Y_axis # OEIS-Other: A000124 planepath=PyramidRows,step=1,align=right line_type=Y_axis }, 'step=1,align=left,n_start=1' => { Diagonal_NW => 'A000124', # triangular+1 = n*(n+1)/2+1 # OEIS-Other: A000124 planepath=PyramidRows,step=1,align=left line_type=Diagonal_NW }, do { my $href = { Y_axis => 'A000217', # triangular }; ('step=1,align=centre,n_start=0' => $href, 'step=1,align=right,n_start=0' => $href); # OEIS-Other: A000217 planepath=PyramidRows,step=1,n_start=0 line_type=Y_axis # OEIS-Other: A000217 planepath=PyramidRows,step=1,align=right,n_start=0 line_type=Y_axis }, 'step=2,align=centre,n_start=1' => { Diagonal_NW => 'A002522', # n^2+1 # OEIS-Other: A002522 planepath=PyramidRows,step=2 line_type=Diagonal_NW # Not quite, n_start=1 means squares starting from 1 whereas A000290 # starts from 0 # Diagonal => 'A000290', }, 'step=2,align=centre,n_start=0' => { Y_axis => 'A002378', # pronic Diagonal => 'A005563', Diagonal_NW => 'A000290', # squares # OEIS-Other: A002378 planepath=PyramidRows,step=2,n_start=0 line_type=Y_axis # OEIS-Other: A005563 planepath=PyramidRows,step=2,n_start=0 line_type=Diagonal # OEIS-Other: A000290 planepath=PyramidRows,step=2,n_start=0 line_type=Diagonal_NW }, 'step=2,align=right,n_start=0' => { Y_axis => 'A000290', # squares Diagonal => 'A002378', # pronic # OEIS-Other: A000290 planepath=PyramidRows,step=2,align=right,n_start=0 line_type=Y_axis # OEIS-Other: A002378 planepath=PyramidRows,step=2,align=right,n_start=0 line_type=Diagonal }, 'step=2,align=left,n_start=0' => { Y_axis => 'A005563', Diagonal_NW => 'A002378', # pronic # OEIS-Other: A005563 planepath=PyramidRows,step=2,align=left,n_start=0 line_type=Y_axis # OEIS-Other: A002378 planepath=PyramidRows,step=2,align=left,n_start=0 line_type=Diagonal_NW }, 'step=2,align=centre,n_start=2' => { Diagonal_NW => 'A059100', # n^2+2 # OEIS-Other: A059100 planepath=PyramidRows,step=2,n_start=2 line_type=Diagonal_NW }, 'step=2,align=right,n_start=2' => { Y_axis => 'A059100', # n^2+2 # OEIS-Other: A059100 planepath=PyramidRows,step=2,align=right,n_start=2 line_type=Y_axis }, 'step=3,align=centre,n_start=1' => { Y_axis => 'A104249', Diagonal_NW => 'A143689', # OEIS-Catalogue: A104249 planepath=PyramidRows,step=3 line_type=Y_axis # OEIS-Catalogue: A143689 planepath=PyramidRows,step=3 line_type=Diagonal_NW # Not quite OFFSET=1 cf start i=0 here # Diagonal => 'A005448', # # OEIS-Catalogue: A005448 planepath=PyramidRows,step=3 line_type=Diagonal }, 'step=3,align=right,n_start=1' => { Y_axis => 'A143689', Diagonal => 'A104249', # OEIS-Other: A143689 planepath=PyramidRows,step=3,align=right line_type=Y_axis # OEIS-Other: A104249 planepath=PyramidRows,step=3,align=right line_type=Diagonal # Not quite OFFSET=1 cf start i=0 here # Diagonal => 'A005448', # # OEIS-Catalogue: A005448 planepath=PyramidRows,step=3 line_type=Diagonal }, 'step=3,align=centre,n_start=0' => { Y_axis => 'A005449', # second pentagonal n*(3n+1)/2 Diagonal_NW => 'A000326', # pentagonal n(3n-1)/2 # OEIS-Other: A005449 planepath=PyramidRows,step=3,n_start=0 line_type=Y_axis # OEIS-Other: A000326 planepath=PyramidRows,step=3,n_start=0 line_type=Diagonal_NW }, 'step=3,align=right,n_start=0' => { Y_axis => 'A000326', # pentagonal n(3n-1)/2 Diagonal => 'A005449', # second pentagonal n*(3n+1)/2 # OEIS-Other: A000326 planepath=PyramidRows,step=3,align=right,n_start=0 line_type=Y_axis # OEIS-Other: A005449 planepath=PyramidRows,step=3,align=right,n_start=0 line_type=Diagonal }, 'step=4,align=centre,n_start=1' => { Y_axis => 'A084849', Diagonal => 'A001844', Diagonal_NW => 'A058331', # OEIS-Catalogue: A084849 planepath=PyramidRows,step=4 line_type=Y_axis # OEIS-Other: A001844 planepath=PyramidRows,step=4 line_type=Diagonal # OEIS-Other: A058331 planepath=PyramidRows,step=4 line_type=Diagonal_NW }, 'step=4,align=right,n_start=1' => { Diagonal => 'A058331', # OEIS-Other: A058331 planepath=PyramidRows,step=4,align=right line_type=Diagonal }, 'step=4,align=left,n_start=1' => { Diagonal_NW => 'A001844', # OEIS-Other: A001844 planepath=PyramidRows,step=4,align=left line_type=Diagonal_NW }, 'step=4,align=centre,n_start=0' => { Y_axis => 'A014105', # second hexagonal Diagonal => 'A046092', # 4*triangular Diagonal_NW => 'A001105', # 2*n^2 # OEIS-Other: A014105 planepath=PyramidRows,step=4,n_start=0 line_type=Y_axis # OEIS-Catalogue: A046092 planepath=PyramidRows,step=4,n_start=0 line_type=Diagonal # OEIS-Other: A001105 planepath=PyramidRows,step=4,n_start=0 line_type=Diagonal_NW }, 'step=4,align=right,n_start=0' => { Y_axis => 'A000384', # 2*n^2-n, hexagonal numbers Diagonal => 'A001105', # OEIS-Other: A000384 planepath=PyramidRows,step=4,align=right,n_start=0 line_type=Y_axis # OEIS-Other: A001105 planepath=PyramidRows,step=4,align=right,n_start=0 line_type=Diagonal }, 'step=4,align=left,n_start=0' => { Diagonal_NW => 'A046092', # 4*triangular # OEIS-Other: A046092 planepath=PyramidRows,step=4,align=left,n_start=0 line_type=Diagonal_NW }, # TODO PyramidRows,step=5 n_start=0 'step=5,align=centre,n_start=1' => { Y_axis => 'A116668', # OEIS-Other: A116668 planepath=PyramidRows,step=5 line_type=Y_axis }, 'step=6,align=centre,n_start=1' => { Diagonal_NW => 'A056107', Y_axis => 'A056108', Diagonal => 'A056109', # OEIS-Other: A056107 planepath=PyramidRows,step=6 line_type=Diagonal_NW # OEIS-Other: A056108 planepath=PyramidRows,step=6 line_type=Y_axis # OEIS-Other: A056109 planepath=PyramidRows,step=6 line_type=Diagonal }, 'step=8,align=centre,n_start=1' => { Diagonal_NW => 'A053755', # OEIS-Other: A053755 planepath=PyramidRows,step=8 line_type=Diagonal_NW }, 'step=9,align=centre,n_start=1' => { Y_axis => 'A006137', Diagonal => 'A038764', # OEIS-Other: A006137 planepath=PyramidRows,step=9 line_type=Y_axis # OEIS-Other: A038764 planepath=PyramidRows,step=9 line_type=Diagonal }, }; } { package Math::PlanePath::PyramidSides; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { X_neg => 'A002522', Diagonal => 'A033951', # OEIS-Catalogue: A002522 planepath=PyramidSides line_type=X_neg # OEIS-Other: A033951 planepath=PyramidSides line_type=Diagonal # # X_axis -- squares (x+1)^2, but starting i=0 value=1 } }; } { package Math::PlanePath::CellularRule; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'rule=5,n_start=1' => { Y_axis => 'A061925', # ceil(n^2/2)+1 # OEIS-Catalogue: A061925 planepath=CellularRule,rule=5 line_type=Y_axis }, # rule 84,116,212,244 two-wide right line do { my $tworight = { Diagonal => 'A005408', # odds 2n+1 }; ('rule=84,n_start=1' => $tworight, 'rule=116,n_start=1' => $tworight, 'rule=212,n_start=1' => $tworight, 'rule=244,n_start=1' => $tworight, ); # OEIS-Other: A005408 planepath=CellularRule,rule=84 line_type=Diagonal # OEIS-Other: A005408 planepath=CellularRule,rule=116 line_type=Diagonal # OEIS-Other: A005408 planepath=CellularRule,rule=212 line_type=Diagonal # OEIS-Other: A005408 planepath=CellularRule,rule=244 line_type=Diagonal }, 'rule=77,n_start=1' => { Y_axis => 'A000124', # triangular+1 # OEIS-Other: A000124 planepath=CellularRule,rule=77 line_type=Y_axis }, 'rule=177,n_start=1' => { Diagonal => 'A000124', # triangular+1 # OEIS-Other: A000124 planepath=CellularRule,rule=177 line_type=Diagonal }, 'rule=185,n_start=1' => { Diagonal => 'A002522', # n^2+1 # OEIS-Other: A002522 planepath=CellularRule,rule=185 line_type=Diagonal }, 'rule=189,n_start=1' => { Y_axis => 'A002522', # n^2+1 # OEIS-Other: A002522 planepath=CellularRule,rule=189 line_type=Y_axis }, # PyramidRows step=1,align=left # OEIS-Other: A000124 planepath=CellularRule,rule=206 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=238 line_type=Diagonal_NW do { my $solidgapright = { Diagonal => 'A002522', # n^2+1 }; ('rule=209,n_start=1' => $solidgapright, 'rule=241,n_start=1' => $solidgapright, ); # OEIS-Other: A002522 planepath=CellularRule,rule=209 line_type=Diagonal # OEIS-Other: A002522 planepath=CellularRule,rule=241 line_type=Diagonal }, 'rule=29,n_start=1' => { Y_axis => 'A000124', # triangular+1 # OEIS-Other: A000124 planepath=CellularRule,rule=29 line_type=Y_axis }, 'rule=221,n_start=1' => { Y_axis => 'A002522', # n^2+1 # OEIS-Other: A002522 planepath=CellularRule,rule=221 line_type=Y_axis }, 'rule=229,n_start=1' => { Y_axis => 'A002522', # n^2+1 # OEIS-Other: A002522 planepath=CellularRule,rule=229 line_type=Y_axis }, # # rule=13 Y axis # # rule=28,156 # Y_axis => 'A002620', quarter squares floor(n^2/4) but diff start # Diagonal => 'A024206', quarter squares - 1, but diff start # # A000027 naturals integers 1 upwards, but OFFSET=1 cf start Y=0 here # # central column only # 'rule=4' => # { Y_axis => 'A000027', # 1 upwards # # OEIS-Other: A000027 planepath=CellularRule,rule=4 line_type=Y_axis # }, # # # right line only rule=16,24,48,56,80,88,112,120,144,152,176,184,208,216,240,248 # Not quite A000027 OFFSET=1 vs start X=Y=0 here # 'rule=16' => # { Y_axis => 'A000027', # 1 upwards # # OEIS-Other: A000027 planepath=CellularRule,rule=16 line_type=Diagonal # }, }; } { package Math::PlanePath::CellularRule::Line; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; } { package Math::PlanePath::CellularRule::OneTwo; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; # use constant _NumSeq_N_oeis_anum => # { 'align=left,n_start=1' => # { # Not quite, OFFSET=1 cf coordinate X=0 here # Diagonal_NW => 'A001651', # not divisible by 3 # }, # { 'align=right,n_start=1' => # Not quite, A032766 0 or 1 mod 3, but it starts OFFSET=0 value=0 # whereas path start 1,3,4,etc without initial 0 # Diagonal => 'A032766', # # }; } { package Math::PlanePath::CellularRule::Two; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'align=right,n_start=1' => { Diagonal => 'A005408', # odd 2n+1 # OEIS-Other: A005408 planepath=CellularRule,rule=84 line_type=Diagonal }, }; } { package Math::PlanePath::CellularRule::OddSolid; # rule=50,58,114,122,178,179,186,242,250 pyramid every second point use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { Diagonal_NW => 'A000124', # triangular+1 # OEIS-Other: A000124 planepath=CellularRule,rule=50 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=58 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=114 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=122 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=178 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=179 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=186 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=242 line_type=Diagonal_NW # OEIS-Other: A000124 planepath=CellularRule,rule=250 line_type=Diagonal_NW # # Not quite, starts value=0 # Diagonal => 'A000217', # triangular numbers but diff start }, }; } { package Math::PlanePath::CellularRule54; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; } { package Math::PlanePath::CellularRule57; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; } { package Math::PlanePath::CellularRule190; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'mirror=0,n_start=0' => { Diagonal_NW => 'A006578', # triangular and quarter square # OEIS-Catalogue: A006578 planepath=CellularRule190,n_start=0 line_type=Diagonal_NW }, 'mirror=1,n_start=0' => { Diagonal_NW => 'A006578', # triangular and quarter square # OEIS-Other: A006578 planepath=CellularRule190,mirror=1,n_start=0 line_type=Diagonal_NW }, }; } { package Math::PlanePath::UlamWarburton; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'parts=4,n_start=0' => { Depth_start => 'A147562', # cells ON after n stages # OEIS-Catalogue: A147562 planepath=UlamWarburton,n_start=0 line_type=Depth_start }, 'parts=2,n_start=0' => { Depth_start => 'A183060', # num cells ON, starting from 0 X_axis => 'A183060', # X_axis == Depth_start # OEIS-Catalogue: A183060 planepath=UlamWarburton,parts=2,n_start=0 line_type=Depth_start # OEIS-Other: A183060 planepath=UlamWarburton,parts=2,n_start=0 }, 'parts=1,n_start=1' => { Depth_end => 'A151922', # num cells ON at end of depth=n Y_axis => 'A151922', # Y_axis == Depth_end # OEIS-Catalogue: A151922 planepath=UlamWarburton,parts=1,n_start=1 line_type=Depth_end # OEIS-Other: A151922 planepath=UlamWarburton,parts=1,n_start=1 line_type=Y_axis }, }; } { package Math::PlanePath::UlamWarburtonQuarter; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # low 10111=23 increment to 11000=24 # 4^2*3+4*3^2+1*3^3 = 111 123 # 4^3*3 = 192 150 +27 = 3^3 use constant _NumSeq_N_oeis_anum => { 'parts=1,n_start=1' => { Depth_end => 'A151920', # 3^count1bits(n), OFFSET=0 1,2,5,6,9 # OEIS-Catalogue: A151920 planepath=UlamWarburtonQuarter line_type=Depth_end }, }; } { package Math::PlanePath::CoprimeColumns; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_X_offset => 1; # CoprimeColumns # X_axis => 'A002088', # cumulative totient but start X=1 value=0; # Diagonal A015614 cumulative-1 but start X=1 value=1 } { package Math::PlanePath::DivisibleColumns; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # Not quite, X_axis => 'A006218' but path start X=1 cf OFFSET=0, # Not quite, Diagonal => 'A077597' but path start X=1 cf OFFSET=0 } # { package Math::PlanePath::File; # # File points from a disk file # # FIXME: analyze points for min/max # } # { package Math::PlanePath::QuintetCurve; # } # { package Math::PlanePath::QuintetCentres; # # inherit QuintetCurve # } { package Math::PlanePath::CornerReplicate; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { '' => { X_axis => 'A000695', # base 4 digits 0,1 only Y_axis => 'A001196', # base 4 digits 0,3 only Diagonal => 'A062880', # base 4 digits 0,2 only # OEIS-Other: A000695 planepath=CornerReplicate # OEIS-Other: A001196 planepath=CornerReplicate line_type=Y_axis # OEIS-Other: A062880 planepath=CornerReplicate line_type=Diagonal }, }; } { package Math::PlanePath::DigitGroups; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'radix=2,i_start=1' => { X_axis => 'A084471', # 0 -> 00 in binary, starting OFFSET=1 # OEIS-Catalogue: A084471 planepath=DigitGroups,radix=2 i_start=1 }, }; } { package Math::PlanePath::FibonacciWordFractal; use constant _NumSeq_X_axis_increasing => 1; # when touched use constant _NumSeq_Y_axis_increasing => 1; # when touched use constant _NumSeq_Diagonal_increasing => 1; # when touched } { package Math::PlanePath::LTiling; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'L_fill=middle' => { Diagonal => 'A062880', # base 4 digits 0,2 only # OEIS-Other: A062880 planepath=LTiling line_type=Diagonal }, }; } { package Math::PlanePath::WythoffArray; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_N_oeis_anum => { 'x_start=1,y_start=1' => { Y_axis => 'A003622', # spectrum of phi 1,4,6,9 Diagonal => 'A020941', # diagonal, OFFSET=1 # OEIS-Catalogue: A003622 planepath=WythoffArray,x_start=1,y_start=1 line_type=Y_axis # OEIS-Catalogue: A020941 planepath=WythoffArray,x_start=1,y_start=1 line_type=Diagonal # Y=1 every second => 'A005248', # every second Lucas number }, # # Not quite, extra initial 0,1 in A000045 Fibonaccis # # X_axis => 'A000045', # # # Y=1 row X=0,2,4,etc => 'A005248', # every second Lucas number }; } { package Math::PlanePath::WythoffPreliminaryTriangle; use constant _NumSeq_Y_axis_i_start => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_N_oeis_anum => { '' => { Y_axis => 'A173027', # OEIS-Other: A173027 planepath=WythoffPreliminaryTriangle line_type=Y_axis }, }; } { package Math::PlanePath::PowerArray; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # cf Not quite A168183 non-multiples-of-9, A168186 non-multiples-of-12 # are values on Y axis, except OFFSET=1 value=1, whereas path start Y=0 # value=1 use constant _NumSeq_N_oeis_anum => { 'radix=2' => { X_axis => 'A000079', # powers 2^X Y_axis => 'A005408', # odd 2n+1 Diagonal => 'A014480', # (2n+1)*2^n starting n=0 # OEIS-Other: A000079 planepath=PowerArray # OEIS-Other: A005408 planepath=PowerArray line_type=Y_axis # OEIS-Catalogue: A014480 planepath=PowerArray line_type=Diagonal }, 'radix=3' => { X_axis => 'A000244', # powers 3^X # OEIS-Other: A000244 planepath=PowerArray,radix=3 # # Not quite, OFFSET=1 cf path start Y=0 # Y_axis => 'A001651', # non multiples of 3 }, 'radix=4' => { X_axis => 'A000302', # powers 4^X # OEIS-Other: A000302 planepath=PowerArray,radix=4 }, 'radix=5' => { X_axis => 'A000351', # powers 5^X # OEIS-Other: A000351 planepath=PowerArray,radix=5 }, 'radix=10' => { X_axis => 'A011557', # powers 10^X # OEIS-Other: A011557 planepath=PowerArray,radix=10 # Not quite, A067251 OFFSET=1 value=1 whereas path Y=0 N=value=1 # Y_axis => 'A067251', # no trailing 0 digits # # OEIS-Catalogue: A067251 planepath=PowerArray,radix=10 line_type=Y_axis }, }; } #------------------------------------------------------------------------------ # Math-PlanePath-Toothpick { package Math::PlanePath::ToothpickTree; # X axis has N=0 or N=0,1 only, except for parts=octant axis at Y=1 sub _NumSeq_X_axis_increasing { my ($self) = @_; return ($self->{'parts'} ne 'octant'); } # Y axis has N=0,N=1 only, except for parts=octant_up axis at X=1 sub _NumSeq_Y_axis_increasing { my ($self) = @_; return ($self->{'parts'} ne 'octant_up'); } use constant _NumSeq_Y_neg_increasing => 1; # N=0,N=2 only use constant _NumSeq_Diagonal_increasing => 1; # growth steps along diags use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; # catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick use constant _NumSeq_N_oeis_anum => { 'parts=4' => { Depth_start => 'A139250', # OEIS-Other: A139250 planepath=ToothpickTree,parts=4 line_type=Depth_start }, 'parts=3' => { Depth_start => 'A153006', # OEIS-Other: A153006 planepath=ToothpickTree,parts=3 line_type=Depth_start }, 'parts=2' => { Depth_start => 'A152998', # OEIS-Other: A152998 planepath=ToothpickTree,parts=2 line_type=Depth_start }, 'parts=1' => { Depth_start => 'A153000', # OEIS-Other: A153000 planepath=ToothpickTree,parts=1 line_type=Depth_start }, 'parts=wedge' => { Depth_start => 'A160406', # OEIS-Other: A160406 planepath=ToothpickTree,parts=wedge line_type=Depth_start }, 'parts=two_horiz' => { Depth_start => 'A160158', # OEIS-Other: A160158 planepath=ToothpickTree,parts=two_horiz line_type=Depth_start }, }; } { package Math::PlanePath::ToothpickReplicate; use constant _NumSeq_Y_axis_increasing => 1; # N=0,N=1 only sub _NumSeq_Y_neg_increasing { my ($self) = @_; return ($self->{'parts'} == 3 ? 0 # replication twist : 1); # N=0,N=2 only } use constant _NumSeq_Diagonal_increasing => 1; # replicate along diags use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; } { package Math::PlanePath::ToothpickUpist; use constant _NumSeq_Y_axis_increasing => 1; # rows increasing use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; # catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick use constant _NumSeq_N_oeis_anum => { '' => { Depth_start => 'A151566', # OEIS-Other: A151566 planepath=ToothpickUpist line_type=Depth_start }, }; } { package Math::PlanePath::ToothpickSpiral; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; # catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick use constant _NumSeq_N_oeis_anum => { 'n_start=1' => { Diagonal => 'A014634', # odd-index hexagonals Diagonal_NW => 'A033567', # Diagonal_SW => 'A185438', # Diagonal_SE => 'A188135', # # OEIS-Other: A014634 planepath=ToothpickSpiral line_type=Diagonal # OEIS-Other: A033567 planepath=ToothpickSpiral line_type=Diagonal_NW # OEIS-Other: A185438 planepath=ToothpickSpiral line_type=Diagonal_SW # OEIS-Other: A188135 planepath=ToothpickSpiral line_type=Diagonal_SE }, 'n_start=0' => { Diagonal => 'A033587', # Diagonal_SW => 'A014635', # even-index hexagonals Diagonal_SE => 'A033585', # Diagonal_NW => 'A139271', # OEIS-Other: A033587 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal # OEIS-Other: A014635 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal_SW # OEIS-Other: A033585 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal_SE # OEIS-Other: A139271 planepath=ToothpickSpiral,n_start=0 line_type=Diagonal_NW }, }; } { package Math::PlanePath::LCornerTree; sub _NumSeq_X_axis_increasing { my ($self) = @_; return $self->{'parts'} ne 'diagonal-1'; } { my %_NumSeq_Y_axis_increasing = ('octant+1' => 1, # two points only 'diagonal-1' => 1, ); sub _NumSeq_Y_axis_increasing { my ($self) = @_; return $_NumSeq_Y_axis_increasing{$self->{'parts'}}; } } sub _NumSeq_X_neg_increasing { my ($self) = @_; return ($self->{'parts'} eq 'wedge' # two points N=0,N=1 || $self->{'parts'} eq 'wedge+1' # three points N=0,1,7 || $self->{'parts'} eq 'diagonal'); } # parts=diagonal has minimum N=0 at X=0,Y=-1, so explicit Y_neg minimum use constant _NumSeq_Y_neg_min => 0; sub _NumSeq_Y_neg_increasing { my ($self) = @_; return $self->{'parts'} ne 'diagonal'; } use constant _NumSeq_Diagonal_increasing => 1; # growth along diags use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; # catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick use constant _NumSeq_N_oeis_anum => { 'parts=4' => { Depth_start => 'A160410', # 4 * cumulative 3^count1bits(n) # OEIS-Other: A160410 planepath=LCornerTree line_type=Depth_start }, 'parts=3' => { Depth_start => 'A160412', # 3 * cumulative 3^count1bits(n) # OEIS-Other: A160412 planepath=LCornerTree,parts=3 line_type=Depth_start }, 'parts=diagonal-1' => { Depth_start => 'A183148', # half-plane triplet toothpicks # OEIS-Other: A183148 planepath=LCornerTree,parts=diagonal-1 line_type=Depth_start # No, N=1 start of depth=1 is at X=1,Y=0 not on SE diagonal # Diagonal_SE => 'A183148', }, # Not quite, A130665=1,4,7,16 offset=0 whereas Nend=0,1,4,7,16 depth=0 # has extra initial 0. # 'parts=1' => # { Depth_end => 'A130665', # cumulative 3^count1bits(d), starting a(0)=1 # # OEIS-Catalogue: A130665 planepath=LCornerTree,parts=1,n_start=-1 line_type=Depth_end i_offset=1 # }, }; } { package Math::PlanePath::LCornerReplicate; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; # replicate along diags # catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick use constant _NumSeq_N_oeis_anum => { '' => { Diagonal => 'A062880', # base 4 digits 0,2 only # OEIS-Other: A062880 planepath=LCornerReplicate line_type=Diagonal }, }; } { package Math::PlanePath::OneOfEight; use constant _NumSeq_X_axis_increasing => 1; use constant _NumSeq_X_neg_increasing => 1; use constant _NumSeq_Y_axis_increasing => 1; use constant _NumSeq_Y_neg_increasing => 1; use constant _NumSeq_Diagonal_increasing => 1; use constant _NumSeq_Diagonal_NW_increasing => 1; use constant _NumSeq_Diagonal_SW_increasing => 1; use constant _NumSeq_Diagonal_SE_increasing => 1; # catalogued in Math::NumSeq::OEIS::Catalogue::Plugin::PlanePathToothpick use constant _NumSeq_N_oeis_anum => { 'parts=4' => { Depth_start => 'A151725', # OEIS-Other: A151725 planepath=OneOfEight line_type=Depth_start }, 'parts=1' => { Depth_start => 'A151735', # OEIS-Other: A151735 planepath=OneOfEight,parts=1 line_type=Depth_start }, 'parts=3mid' => { Depth_start => 'A170880', # OEIS-Other: A170880 planepath=OneOfEight,parts=3mid line_type=Depth_start }, 'parts=3side' => { Depth_start => 'A170879', # OEIS-Other: A170879 planepath=OneOfEight,parts=3side line_type=Depth_start }, }; } { package Math::PlanePath::HTree; # clockwise around each sub-tree so N increases along X axis use constant _NumSeq_X_axis_increasing => 1; } #------------------------------------------------------------------------------ { package Math::PlanePath; use constant _NumSeq_A2 => 0; } { package Math::PlanePath::TriangleSpiral; use constant _NumSeq_A2 => 1; } { package Math::PlanePath::HexSpiral; use constant _NumSeq_A2 => 1; } { package Math::PlanePath::HexArms; use constant _NumSeq_A2 => 1; } { package Math::PlanePath::TriangularHypot; use constant _NumSeq_A2 => 1; } { package Math::PlanePath::Flowsnake; use constant _NumSeq_A2 => 1; # and FlowsnakeCentres inherits } 1; __END__ =for stopwords Ryde Math-PlanePath SquareSpiral lookup PlanePath ie =head1 NAME Math::NumSeq::PlanePathN -- sequence of N values from PlanePath module =head1 SYNOPSIS use Math::NumSeq::PlanePathN; my $seq = Math::NumSeq::PlanePathN->new (planepath => 'SquareSpiral', line_type => 'X_axis'); my ($i, $value) = $seq->next; =head1 DESCRIPTION This module presents N values from a C as a sequence. The default is the X axis, or the C parameter (a string) can choose among "X_axis" X axis (positive part) "Y_axis" Y axis (positive part) "X_neg" X negative axis "Y_neg" Y negative axis "Diagonal" leading diagonal X=i, Y=i "Diagonal_NW" north-west diagonal X=-i, Y=i "Diagonal_SW" south-west diagonal X=-i, Y=-i "Diagonal_SE" south-east diagonal X=i, Y=-i "Depth_start" first N at depth=i "Depth_end" last N at depth=i For example the C X axis starts i=0 with values 1, 2, 11, 28, 53, 86, etc. "X_neg", "Y_neg", "Diagonal_NW", etc, on paths which don't traverse negative X or Y have just a single value from X=0,Y=0. The behaviour on paths which visit only some of the points on the respective axis is unspecified as yet, as is behaviour on paths with repeat points, such as the C. =head1 FUNCTIONS See L for behaviour common to all sequence classes. =over 4 =item C<$seq = Math::NumSeq::PlanePathN-Enew (key=Evalue,...)> Create and return a new sequence object. The options are planepath string, name of a PlanePath module planepath_object PlanePath object line_type string, as described above C can be either the module part such as "SquareSpiral" or a full class name "Math::PlanePath::SquareSpiral". =item C<$value = $seq-Eith($i)> Return the N value at C<$i> in the PlanePath. C<$i> gives a position on the respective C, so the X,Y to lookup a C<$value=N> is X,Y line_type ----- --------- $i, 0 "X_axis" 0, $i "Y_axis" -$i, 0 "X_neg" 0, -$i "Y_neg" $i, $i "Diagonal" $i, -$i "Diagonal_NW" -$i, -$i "Diagonal_SW" $i, -$i "Diagonal_SE" =item C<$bool = $seq-Epred($value)> Return true if C<$value> occurs in the sequence. This means C<$value> is an integer N which is on the respective C, ie. that C<($path-En_to_xy($value)> is on the line type. =back =head1 SEE ALSO L, L, L =head1 HOME PAGE L =head1 LICENSE Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020, 2021 Kevin Ryde This file is part of Math-PlanePath. Math-PlanePath is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version. Math-PlanePath is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with Math-PlanePath. If not, see . =cut