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Math::PlanePath::PyramidSides -- points along the sides of pyramid
use Math::PlanePath::PyramidSides; my $path = Math::PlanePath::PyramidSides->new; my ($x, $y) = $path->n_to_xy (123);
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.
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
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
$non the path.
$n < 0.5the 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
$yare each rounded to the nearest integer which has the effect of treating points in the pyramid as a squares of side 1, so the half-plane y>=-0.5 is entirely covered.
($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)
The returned range is exact, meaning
$n_hiare the smallest and biggest in the rectangle.
Rectangle to N Range
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.
$x2 have different signs, then X=0 is the smallest.
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), floor(sqrt(N)), k repeated 2k+1 times starting 0
Copyright 2010, 2011, 2012, 2013, 2014 Kevin Ryde
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