NAME
Math::PlanePath::MPeaks  points in expanding M shape
SYNOPSIS
use Math::PlanePath::MPeaks;
my $path = Math::PlanePath::MPeaks>new;
my ($x, $y) = $path>n_to_xy (123);
DESCRIPTION
This path puts points in layers of an "M" shape
41 49 7
40 42 48 50 6
39 22 43 47 28 51 5
38 21 23 44 46 27 29 52 4
37 20 9 24 45 26 13 30 53 3
36 19 8 10 25 12 14 31 54 2
35 18 7 2 11 4 15 32 55 1
34 17 6 1 3 5 16 33 56 < Y=0
^
4 3 2 1 X=0 1 2 3 4
N=1 to N=5 is the first "M" shape, then N=6 to N=16 on top of that, etc. The centre goes half way down. Reckoning the N=1 to N=5 as layer d=1 then
Xleft = d
Xright = d
Ypeak = 2*d  1
Ycentre = d  1
Each "M" is 6 points longer than the preceding. The verticals are each 2 longer, and the centre diagonals each 1 longer. This step 6 is similar to the HexSpiral
.
The octagonal numbers N=1,8,21,40,65,etc k*(3k2) are a straight line of slope 2 going up to the left. The octagonal numbers of the second kind N=5,16,33,56,etc k*(3k+2) are along the X axis to the right.
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,
n_start => 0
40 48
39 41 47 49
38 21 42 46 27 50
37 20 22 43 45 26 28 51
36 19 8 23 44 25 12 29 52
35 18 7 9 24 11 13 30 53
34 17 6 1 10 3 14 31 54
33 16 5 0 2 4 15 32 55
FUNCTIONS
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::MPeaks>new ()

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

Return the X,Y coordinates of point number
$n
on the path.For
$n < 0.5
the return is an empty list, it being considered there are no negative points. $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 points as a squares of side 1, so the halfplane y>=0.5 is entirely covered.
OEIS
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A045944 (etc)
n_start=1 (the default)
A045944 N on X axis >= 1, extra initial 0
being octagonal numbers second kind
A056106 N on Y axis, extra initial 1
A056109 N on X negative axis <= 1
n_start=0
A049450 N on Y axis, extra initial 0, 2*pentagonal
n_start=2
A027599 N on Y axis, extra initial 6,2
SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidSides
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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/>.