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# 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

``````    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``````

http://user42.tuxfamily.org/math-planepath/index.html