NAME
Math::PlanePath::PentSpiralSkewed  integer points in a pentagonal shape
SYNOPSIS
use Math::PlanePath::PentSpiralSkewed;
my $path = Math::PlanePath::PentSpiralSkewed>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path makes a pentagonal (fivesided) spiral with points skewed so as to fit a square grid and fully cover the plane.
10 ... 2
/ \ \
11 3 9 20 1
/ / \ \ \
12 4 12 8 19 < Y=0
\ \  
13 567 18 1
\ 
14151617 2
^ ^ ^ ^ ^ ^
2 1 X=0 1 2 3 ...
The pattern is similar to the SquareSpiral
but cuts three corners which makes each cycle is faster. Each cycle is just 5 steps longer than the previous (where it's 8 for a SquareSpiral
).
N Start
The default is to number points starting N=1 as shown above. An optional n_start
can give a different start, in the same pattern. For example to start at 0,
38 n_start => 0
39 21 37 ...
40 22 9 20 36 57
41 23 10 2 8 19 35 56
42 24 11 3 0 1 7 18 34 55
43 25 12 4 5 6 17 33 54
44 26 13 14 15 16 32 53
45 27 28 29 30 31 52
46 47 48 49 50 51
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::PentSpiral>new ()
$path = Math::PlanePath::PentSpiral>new (n_start => $n)

Create and return a new path object.
$n = $path>xy_to_n ($x,$y)

Return the point number for coordinates
$x,$y
.$x
and$y
are each rounded to the nearest integer, which has the effect of treating each point in the path as a square of side 1.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A192136 (etc)
n_start=1 (the default)
A192136 N on X axis, (5*n^2  3*n + 2)/2
A140066 N on Y axis
A116668 N on X negative axis, (5n^2 + n + 2)/2
A134238 N on Y negative axis
A158187 N on NorthWest diagonal, 10*n^2 + 1
A005891 N on SouthEast diagonal, centred pentagonals
n_start=0
A000566 N on X axis, heptagonal numbers
A005476 N on Y axis
A005475 N on X negative axis
A147875 N on Y negative axis, second heptagonals
A033583 N on NorthWest diagonal, 10*n^2
A028895 N on SouthEast diagonal, 5*triangular
SEE ALSO
Math::PlanePath, Math::PlanePath::SquareSpiral, Math::PlanePath::DiamondSpiral, Math::PlanePath::HexSpiralSkewed
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019 Kevin Ryde
This file is part of MathPlanePath.
MathPlanePath 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.
MathPlanePath 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 MathPlanePath. If not, see <http://www.gnu.org/licenses/>.