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 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.
A step
parameter controls how much wider each row is than the preceding, to make wider pyramids. For example step 4
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.
step => 1 11 12 13 14 15 4 7 8 9 10 3 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 unlikely to be much use. The Rows path with width
1 does this too.
step => 0 5 4 4 3 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".
"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 => 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.
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.
For $n <= 0
the return is an empty list since the path starts at N=1.
$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
.
($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.
$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 Kevin Ryde
This file is part of Math-PlanePath.
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