Math::PlanePath::PyramidRows -- points stacked up in a pyramid
use Math::PlanePath::PyramidRows; my $path = Math::PlanePath::PyramidRows->new; my ($x, $y) = $path->n_to_xy (123);
This path arranges points in successively wider rows going upwards so as to form an upside-down pyramid. The default step is 2, ie. each row 2 wider than the preceding, an extra point at the left and the right,
17 18 19 20 21 22 23 24 25 4 10 11 12 13 14 15 16 3 step => 2 5 6 7 8 9 2 2 3 4 1 1 <- Y=0 -4 -3 -2 -1 X=0 1 2 3 4 ...
The right end N=1,4,9,16,etc is the perfect squares. The vertical 2,6,12,20,etc at x=-1 is the pronic numbers s*(s+1), half way between those successive squares.
The step 2 is the same as the PyramidSides, Corner and SacksSpiral paths. For the SacksSpiral, spiral arms going to the right correspond to diagonals in the pyramid, and arms to the left correspond to verticals.
PyramidSides
Corner
SacksSpiral
A step parameter controls how much wider each row is than the preceding, to make wider pyramids. For example step 4
step
my $path = Math::PlanePath::PyramidRows->new (step => 4);
makes each row 2 wider on each side successively
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 4 16 17 18 19 20 21 22 23 24 25 26 27 28 3 7 8 9 10 11 12 13 14 15 2 2 3 4 5 6 1 1 <- Y=0 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 ...
If the step is an odd number then the extra is at the right, so step 3 gives
13 14 15 16 17 18 19 20 21 22 3 6 7 8 9 10 11 12 2 2 3 4 5 1 1 <- Y=0 -3 -2 -1 X=0 1 2 3 4 ...
Or step 1 goes solely to the right. This is equivalent to the Diagonals path, but columns shifted up to make horizontal rows.
11 12 13 14 15 4 7 8 9 10 3 step => 1 4 5 6 2 2 3 1 1 <- Y=0 X=0 1 2 3 4 ...
Step 0 means simply a vertical, each row 1 wide and not increasing. This is consistent but unlikely to be much use. The Rows path with width 1 does this too.
width
5 4 4 3 step => 0 3 2 2 1 1 <- Y=0 X=0
Various number sequences fall in regular patterns positions depending on the step. Large steps are not particularly interesting and quickly become very wide. A limit might be desirable in a user interface, but there's no limit in the code as such.
An optional align parameter controls how the points are arranged relative to the Y axis. The default shown above is "centre".
align
"right" means points to the right of the axis,
align=>"right" 26 27 28 29 30 31 32 33 34 35 36 5 17 18 19 20 21 22 23 24 25 4 10 11 12 13 14 15 16 3 5 6 7 8 9 2 2 3 4 1 1 <- Y=0 X=0 1 2 3 4 5 6 7 8 9 10
"left" is similar but to the left of the Y axis, ie. into negative X.
align=>"left" 26 27 28 29 30 31 32 33 34 35 36 5 17 18 19 20 21 22 23 24 25 4 10 11 12 13 14 15 16 3 5 6 7 8 9 2 2 3 4 1 1 <- Y=0 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0
The step parameter still controls how much longer each row is than its predecessor.
The default is to number points starting N=1 as shown above. An optional n_start can give a different start, in the same rows sequence. For example to start at 0,
n_start
n_start => 0 16 17 18 19 20 21 22 23 24 4 9 10 11 12 13 14 15 3 4 5 6 7 8 2 1 2 3 1 0 <- Y=0 -------------------------- -4 -3 -2 -1 X=0 1 2 3 4
For step=3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2, are at the rightmost end of each row. The second pentagonal numbers 2,7,15,26, S(k) = (3k+1)*k/2 are the vertical at x=-1. Those second numbers are obtained by P(-k), and the two together are the "generalized pentagonal numbers".
Both these sequences are composites from 12 and 15 onwards, respectively, and the immediately preceding P(k)-1, P(k)-2, and S(k)-1, S(k)-2 are too. They factorize simply as
P(k) = (3*k-1)*k/2 P(k)-1 = (3*k+2)*(k-1)/2 P(k)-2 = (3*k-4)*(k-1)/2 S(k) = (3*k+1)*k/2 S(k)-1 = (3*k-2)*(k+1)/2 S(k)-2 = (3*k+4)*(k-1)/2
Plotting the primes on a step=3 PyramidRows has the second pentagonal S(k),S(k)-1,S(k)-2 as a 3-wide vertical gap of no primes at X=-1,-2,-3. The the plain pentagonal P(k),P(k-1),P(k)-2 are the endmost three N of each row non-prime. The vertical is much more noticeable in a plot.
PyramidRows
no primes these three columns no primes these end three except the low 2,7,13 except low 3,5,11 | | | / / / 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 23 24 25 26 27 28 29 30 31 32 33 34 35 13 14 15 16 17 18 19 20 21 22 6 7 8 9 10 11 12 2 3 4 5 1 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 10 11 ...
With align="left" the end values can be put into columns,
no primes these end three align => "left" except low 3,5,11 | | | 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 5 23 24 25 26 27 28 29 30 31 32 33 34 35 4 13 14 15 16 17 18 19 20 21 22 3 6 7 8 9 10 11 12 2 2 3 4 5 1 1 <- Y=0 ... -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0
In general a constant offset S(k)-c is a column and from P(k)-c is a diagonal sloping up dX=2,dY=1 right. The simple factorizations above using the roots of the quadratic P(k)-c or S(k)-c is possible whenever 24*c+1 is a perfect square. This means the further columns S(k)-5, S(k)-7, S(k)-12, etc also have no primes.
The columns S(k), S(k)-1, S(k)-2 are prominent because they're adjacent. There's no other adjacent columns of this type because the squares after 49 are too far apart for 24*c+1 to be a square for successive c. Of course there could be other reasons for other columns or diagonals to have few or many primes.
See "FUNCTIONS" in Math::PlanePath for behaviour common to all path classes.
$path = Math::PlanePath::PyramidRows->new ()
$path = Math::PlanePath::PyramidRows->new (step => $integer, align => $str, n_start => $n)
Create and return a new path object. The default step is 2. align is a string, one of
"centre" the default "right" points aligned right of the Y axis "left" points aligned left of the Y axis
Points are always numbered from left to right in the rows, the alignment changes where each row begins (or ends).
($x,$y) = $path->n_to_xy ($n)
Return the X,Y coordinates of point number $n on the path.
$n
For $n <= 0 the return is an empty list since the path starts at N=1.
$n <= 0
$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 each point in the pyramid as a square of side 1. If $x,$y is outside the pyramid the return is undef.
$x,$y
$x
$y
undef
($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.
$n_lo
$n_hi
$x = $path->sumxy_minimum()
$x = $path->sumxy_maximum()
Return the minimum or maximum values taken by coordinate sum X+Y reached by integer N values in the path. If there's no minimum or maximum then return undef.
The path is right and above the X=-Y diagonal, thus giving a minimum sum, in the following cases.
align condition for sumxy_minimum=0 ------ ----------------------------- centre step <= 3 right always left step <= 1
$x = $path->diffxy_minimum()
$x = $path->diffxy_maximum()
Return the minimum or maximum values taken by coordinate difference X-Y reached by integer N values in the path. If there's no minimum or maximum then return undef.
The path is left and above the X=Y leading diagonal, thus giving a minimum X-Y difference, in the following cases.
align condition for diffxy_minimum=0 ------ ----------------------------- centre step <= 2 right step <= 1 left always
Entries in Sloane's Online Encyclopedia of Integer Sequences related to this path include
http://oeis.org/A023531 (etc)
step=1 A002262 X coordinate, runs 0 to k A003056 Y coordinate, k repeated k+1 times A051162 X+Y sum A025581 Y-X diff, runs k to 0 A079904 X*Y product A069011 X^2+Y^2, n_to_rsquared() A080099 X bitwise-AND Y A080098 X bitwise-OR Y A051933 X bitwise-XOR Y A050873 GCD(X+1,Y+1) greatest common divisor by rows A051173 LCM(X+1,Y+1) least common multiple by rows A023531 dY, being 1 at triangular numbers (but starting n=0) A167407 dX-dY, change in X-Y (extra initial 0) A129184 turn 1=left, 0=right or straight A079824 N total along each opposite diagonal A000124 N on Y axis (triangular+1) A000217 N on X=Y diagonal, extra initial 0 step=1, n_start=0 A109004 GCD(X,Y) greatest common divisor starting (0,0) A103451 turn 1=left or right,0=straight, but extra initial 1 A103452 turn 1=left,0=straight,-1=right, but extra initial 1 step=2 A196199 X coordinate, runs -n to +n A000196 Y coordinate, n appears 2n+1 times A053186 X+Y, being distance to next higher square A010052 dY, being 1 at perfect square row end A000290 N on X=Y diagonal, extra initial 0 A002522 N on X=-Y North-West diagonal (start row), Y^2+1 A004201 N for which X>=0, ie. right hand half A020703 permutation N at -X,Y step=2, n_start=0 A005563 N on X=Y diagonal, Y*(Y+2) A000290 N on X=-Y North-West diagonal (start row), Y^2 step=2, n_start=2 A059100 N on north-west diagonal (start each row), Y^2+2 A053615 abs(X), runs k..0..k step=2, align=right, n_start=0 A196199 X-Y, runs -k to +k A053615 abs(X-Y), runs k..0..k step=2, align=left, n_start=0 A005563 N on Y axis, Y*(Y+2) step=3 A180447 Y coordinate, n appears 3n+1 times A104249 N on Y axis, Y*(3Y+1)/2+1 A143689 N on X=-Y North-West diagonal step=3, n_start=0 A005449 N on Y axis, second pentagonals Y*(3Y+1)/2 A000326 N on diagonal north-west, pentagonals Y*(3Y-1)/2 step=4 A084849 N on Y axis A001844 N on X=Y diagonal (North-East) A058331 N on X=-Y North-West diagonal A221217 permutation N at -X,Y step=4, n_start=0 A014105 N on Y axis, the second hexagonal numbers A046092 N on X=Y diagonal, 4*triangular numbers step=4, align=right, n_start=0 A060511 X coordinate, amount n exceeds hexagonal number A000384 N on Y axis, the hexagonal numbers A001105 N on X=Y diagonal, 2*squares step=5 A116668 N on Y axis step=6 A056108 N on Y axis A056109 N on X=Y diagonal (North-East) A056107 N on X=-Y North-West diagonal step=8 A053755 N on X=-Y North-West diagonal step=9 A006137 N on Y axis A038764 N on X=Y diagonal (North-East)
Math::PlanePath, Math::PlanePath::PyramidSides, Math::PlanePath::Corner, Math::PlanePath::SacksSpiral, Math::PlanePath::MultipleRings
Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant, Math::PlanePath::Rows
http://user42.tuxfamily.org/math-planepath/index.html
Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019, 2020 Kevin Ryde
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CPAN shell
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