# NAME

Math::PlanePath::PyramidSides -- points along the sides of pyramid

# SYNOPSIS

`````` use Math::PlanePath::PyramidSides;
my \$path = Math::PlanePath::PyramidSides->new;
my (\$x, \$y) = \$path->n_to_xy (123);``````

# DESCRIPTION

This path puts points in layers along the sides of a pyramid growing upwards.

``````                        21                        4
20  13  22                    3
19  12   7  14  23                2
18  11   6   3   8  15  24            1
17  10   5   2   1   4   9  16  25    <- Y=0
------------------------------------
^
... -4  -3  -2  -1  X=0  1   2   3   4 ...``````

N=1,4,9,16,etc along the positive X axis is the perfect squares. N=2,6,12,20,etc in the X=-1 vertical is the pronic numbers k*(k+1) half way between those successive squares.

The pattern is the same as the `Corner` path but turned and spread so the single quadrant in the `Corner` becomes a half-plane here.

The pattern is similar to `PyramidRows` (with its default step=2), just with the columns dropped down vertically to start at the X axis. Any pattern occurring within a column is unchanged, but what was a row becomes a diagonal and vice versa.

## Lucky Numbers of Euler

An interesting sequence for this path is Euler's k^2+k+41. The low values are spread around a bit, but from N=1763 (k=41) they're the vertical at X=40. There's quite a few primes in this quadratic and when plotting primes that vertical stands out a little denser than its surrounds (at least for up to the first 2500 or so values). The line shows in other step==2 paths too, but not as clearly. In the `PyramidRows` for instance the beginning is up at Y=40, and in the `Corner` path it's a diagonal.

## 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 pyramid pattern. For example to start at 0,

``````    n_start => 0

20                    4
19 12 21                 3
18 11  6 13 22              2
17 10  5  2  7 14 23           1
16  9  4  1  0  3  8 15 24    <- Y=0
--------------------------
-4 -3 -2 -1 X=0 1  2  3  4``````

# FUNCTIONS

See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.

`\$path = Math::PlanePath::PyramidSides->new ()`
`\$path = Math::PlanePath::PyramidSides->new (n_start => \$n)`

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 in the pyramid.

`\$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 in the pyramid as a squares of side 1, so the half-plane y>=-0.5 is entirely covered.

`(\$n_lo, \$n_hi) = \$path->rect_to_n_range (\$x1,\$y1, \$x2,\$y2)`

The returned range is exact, meaning `\$n_lo` and `\$n_hi` are the smallest and biggest in the rectangle.

# FORMULAS

## Rectangle to N Range

For `rect_to_n_range()`, in each column N increases so the biggest N is in the topmost row and and smallest N in the bottom row.

In each row N increases along the sequence X=0,-1,1,-2,2,-3,3, etc. So the biggest N is at the X of biggest absolute value and preferring the positive X=k over the negative X=-k.

The smallest N conversely is at the X of smallest absolute value. If the X range crosses 0, ie. `\$x1` and `\$x2` have different signs, then X=0 is the smallest.

# OEIS

Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include

``````    n_start=1 (the default)
A049240    abs(dY), being 0=horizontal step at N=square
A002522    N on X negative axis, x^2+1
A033951    N on X=Y diagonal, 4d^2+3d+1
A004201    N for which X>=0, ie. right hand half
A020703    permutation N at -X,Y

n_start=0
A196199    X coordinate, runs -n to +n
A053615    abs(X), runs n to 0 to n
A000196    abs(X)+abs(Y), being floor(sqrt(N)),
k repeated 2k+1 times starting 0``````

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