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*(3k-2) 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
           half-plane 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/math-planepath/index.html>

LICENSE

       Copyright 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 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 <http://www.gnu.org/licenses/>.