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 halfplane 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 halfplane 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
http://oeis.org/A196199 (etc)
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
SEE ALSO
Math::PlanePath, Math::PlanePath::PyramidRows, Math::PlanePath::Corner, Math::PlanePath::DiamondSpiral, Math::PlanePath::SacksSpiral, Math::PlanePath::MPeaks
HOME PAGE
http://user42.tuxfamily.org/mathplanepath/index.html
LICENSE
Copyright 2010, 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/>.