1Math::PlanePath::PyramiUdsReorwsC(o3n)tributed Perl DocuMmaetnht:a:tPiloannePath::PyramidRows(3)
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6 Math::PlanePath::PyramidRows -- points stacked up in a pyramid
7
9 use Math::PlanePath::PyramidRows;
10 my $path = Math::PlanePath::PyramidRows->new;
11 my ($x, $y) = $path->n_to_xy (123);
12
14 This path arranges points in successively wider rows going upwards so
15 as to form an upside-down pyramid. The default step is 2, ie. each row
16 2 wider than the preceding, an extra point at the left and the right,
17 17 18 19 20 21 22 23 24 25 4
19 10 11 12 13 14 15 16 3 step => 2
20 5 6 7 8 9 2
21 2 3 4 1
22 1 <- Y=0
23 -4 -3 -2 -1 X=0 1 2 3 4 ...
25 The right end N=1,4,9,16,etc is the perfect squares. The vertical
27 2,6,12,20,etc at x=-1 is the pronic numbers s*(s+1), half way between
28 those successive squares.
29 The step 2 is the same as the "PyramidSides", "Corner" and
31 "SacksSpiral" paths. For the "SacksSpiral", spiral arms going to the
32 right correspond to diagonals in the pyramid, and arms to the left
33 correspond to verticals.
34 Step Parameter
36 A "step" parameter controls how much wider each row is than the
37 preceding, to make wider pyramids. For example step 4
38 my $path = Math::PlanePath::PyramidRows->new (step => 4);
40 makes each row 2 wider on each side successively
42 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 4
44 16 17 18 19 20 21 22 23 24 25 26 27 28 3
45 7 8 9 10 11 12 13 14 15 2
46 2 3 4 5 6 1
47 1 <- Y=0
48 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 ...
50 If the step is an odd number then the extra is at the right, so step 3
52 gives
53 13 14 15 16 17 18 19 20 21 22 3
55 6 7 8 9 10 11 12 2
56 2 3 4 5 1
57 1 <- Y=0
58 -3 -2 -1 X=0 1 2 3 4 ...
60 Or step 1 goes solely to the right. This is equivalent to the
62 Diagonals path, but columns shifted up to make horizontal rows.
63 11 12 13 14 15 4
65 7 8 9 10 3 step => 1
66 4 5 6 2
67 2 3 1
68 1 <- Y=0
69 X=0 1 2 3 4 ...
71 Step 0 means simply a vertical, each row 1 wide and not increasing.
73 This is consistent but unlikely to be much use. The Rows path with
74 "width" 1 does this too.
75 5 4
77 4 3 step => 0
78 3 2
79 2 1
80 1 <- Y=0
81 X=0
83 Various number sequences fall in regular patterns positions depending
85 on the step. Large steps are not particularly interesting and quickly
86 become very wide. A limit might be desirable in a user interface, but
87 there's no limit in the code as such.
88 Align Parameter
90 An optional "align" parameter controls how the points are arranged
91 relative to the Y axis. The default shown above is "centre".
92 "right" means points to the right of the axis,
94 align=>"right"
96 26 27 28 29 30 31 32 33 34 35 36 5
98 17 18 19 20 21 22 23 24 25 4
99 10 11 12 13 14 15 16 3
100 5 6 7 8 9 2
101 2 3 4 1
102 1 <- Y=0
103 X=0 1 2 3 4 5 6 7 8 9 10
105 "left" is similar but to the left of the Y axis, ie. into negative X.
107 align=>"left"
109 26 27 28 29 30 31 32 33 34 35 36 5
111 17 18 19 20 21 22 23 24 25 4
112 10 11 12 13 14 15 16 3
113 5 6 7 8 9 2
114 2 3 4 1
115 1 <- Y=0
116 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0
118 The step parameter still controls how much longer each row is than its
120 predecessor.
121 N Start
123 The default is to number points starting N=1 as shown above. An
124 optional "n_start" can give a different start, in the same rows
125 sequence. For example to start at 0,
126 n_start => 0
128 16 17 18 19 20 21 22 23 24 4
130 9 10 11 12 13 14 15 3
131 4 5 6 7 8 2
132 1 2 3 1
133 0 <- Y=0
134 --------------------------
135 -4 -3 -2 -1 X=0 1 2 3 4
136 Step 3 Pentagonals
138 For step=3 the pentagonal numbers 1,5,12,22,etc, P(k) = (3k-1)*k/2, are
139 at the rightmost end of each row. The second pentagonal numbers
140 2,7,15,26, S(k) = (3k+1)*k/2 are the vertical at x=-1. Those second
141 numbers are obtained by P(-k), and the two together are the
142 "generalized pentagonal numbers".
143 Both these sequences are composites from 12 and 15 onwards,
145 respectively, and the immediately preceding P(k)-1, P(k)-2, and S(k)-1,
146 S(k)-2 are too. They factorize simply as
147 P(k) = (3*k-1)*k/2
149 P(k)-1 = (3*k+2)*(k-1)/2
150 P(k)-2 = (3*k-4)*(k-1)/2
151 S(k) = (3*k+1)*k/2
152 S(k)-1 = (3*k-2)*(k+1)/2
153 S(k)-2 = (3*k+4)*(k-1)/2
154 Plotting the primes on a step=3 "PyramidRows" has the second pentagonal
156 S(k),S(k)-1,S(k)-2 as a 3-wide vertical gap of no primes at X=-1,-2,-3.
157 The the plain pentagonal P(k),P(k-1),P(k)-2 are the endmost three N of
158 each row non-prime. The vertical is much more noticeable in a plot.
159 no primes these three columns no primes these end three
161 except the low 2,7,13 except low 3,5,11
162 | | | / / /
163 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70
164 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51
165 23 24 25 26 27 28 29 30 31 32 33 34 35
166 13 14 15 16 17 18 19 20 21 22
167 6 7 8 9 10 11 12
168 2 3 4 5
169 1
170 -6 -5 -4 -3 -2 -1 X=0 1 2 3 4 5 6 7 8 9 10 11 ...
171 With align="left" the end values can be put into columns,
173 no primes these end three
175 align => "left" except low 3,5,11
176 | | |
177 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 5
178 23 24 25 26 27 28 29 30 31 32 33 34 35 4
179 13 14 15 16 17 18 19 20 21 22 3
180 6 7 8 9 10 11 12 2
181 2 3 4 5 1
182 1 <- Y=0
183 ... -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 X=0
184 In general a constant offset S(k)-c is a column and from P(k)-c is a
186 diagonal sloping up dX=2,dY=1 right. The simple factorizations above
187 using the roots of the quadratic P(k)-c or S(k)-c is possible whenever
188 24*c+1 is a perfect square. This means the further columns S(k)-5,
189 S(k)-7, S(k)-12, etc also have no primes.
190 The columns S(k), S(k)-1, S(k)-2 are prominent because they're
192 adjacent. There's no other adjacent columns of this type because the
193 squares after 49 are too far apart for 24*c+1 to be a square for
194 successive c. Of course there could be other reasons for other columns
195 or diagonals to have few or many primes.
196
198 See "FUNCTIONS" in Math::PlanePath for behaviour common to all path
199 classes.
200 "$path = Math::PlanePath::PyramidRows->new ()"
202 "$path = Math::PlanePath::PyramidRows->new (step => $integer, align =>
203 $str, n_start => $n)"
204 Create and return a new path object. The default "step" is 2.
205 "align" is a string, one of
206 "centre" the default
208 "right" points aligned right of the Y axis
209 "left" points aligned left of the Y axis
210 Points are always numbered from left to right in the rows, the
212 alignment changes where each row begins (or ends).
213 "($x,$y) = $path->n_to_xy ($n)"
215 Return the X,Y coordinates of point number $n on the path.
216 For "$n <= 0" the return is an empty list since the path starts at
218 N=1.
219 "$n = $path->xy_to_n ($x,$y)"
221 Return the point number for coordinates "$x,$y". $x and $y are
222 each rounded to the nearest integer, which has the effect of
223 treating each point in the pyramid as a square of side 1. If
224 "$x,$y" is outside the pyramid the return is "undef".
225 "($n_lo, $n_hi) = $path->rect_to_n_range ($x1,$y1, $x2,$y2)"
227 The returned range is exact, meaning $n_lo and $n_hi are the
228 smallest and biggest in the rectangle.
229 Descriptive Methods
231 "$x = $path->sumxy_minimum()"
232 "$x = $path->sumxy_maximum()"
233 Return the minimum or maximum values taken by coordinate sum X+Y
234 reached by integer N values in the path. If there's no minimum or
235 maximum then return "undef".
236 The path is right and above the X=-Y diagonal, thus giving a
238 minimum sum, in the following cases.
239 align condition for sumxy_minimum=0
241 ------ -----------------------------
242 centre step <= 3
243 right always
244 left step <= 1
245 "$x = $path->diffxy_minimum()"
247 "$x = $path->diffxy_maximum()"
248 Return the minimum or maximum values taken by coordinate difference
249 X-Y reached by integer N values in the path. If there's no minimum
250 or maximum then return "undef".
251 The path is left and above the X=Y leading diagonal, thus giving a
253 minimum X-Y difference, in the following cases.
254 align condition for diffxy_minimum=0
256 ------ -----------------------------
257 centre step <= 2
258 right step <= 1
259 left always
260
262 Entries in Sloane's Online Encyclopedia of Integer Sequences related to
263 this path include
264 <http://oeis.org/A023531> (etc)
266 step=1
268 A002262 X coordinate, runs 0 to k
269 A003056 Y coordinate, k repeated k+1 times
270 A051162 X+Y sum
271 A025581 Y-X diff, runs k to 0
272 A079904 X*Y product
273 A069011 X^2+Y^2, n_to_rsquared()
274 A080099 X bitwise-AND Y
275 A080098 X bitwise-OR Y
276 A051933 X bitwise-XOR Y
277 A050873 GCD(X+1,Y+1) greatest common divisor by rows
278 A051173 LCM(X+1,Y+1) least common multiple by rows
279 A023531 dY, being 1 at triangular numbers (but starting n=0)
281 A167407 dX-dY, change in X-Y (extra initial 0)
282 A129184 turn 1=left, 0=right or straight
283 A079824 N total along each opposite diagonal
285 A000124 N on Y axis (triangular+1)
286 A000217 N on X=Y diagonal, extra initial 0
287 step=1, n_start=0
288 A109004 GCD(X,Y) greatest common divisor starting (0,0)
289 A103451 turn 1=left or right,0=straight, but extra initial 1
290 A103452 turn 1=left,0=straight,-1=right, but extra initial 1
291 step=2
293 A196199 X coordinate, runs -n to +n
294 A000196 Y coordinate, n appears 2n+1 times
295 A053186 X+Y, being distance to next higher square
296 A010052 dY, being 1 at perfect square row end
297 A000290 N on X=Y diagonal, extra initial 0
298 A002522 N on X=-Y North-West diagonal (start row), Y^2+1
299 A004201 N for which X>=0, ie. right hand half
300 A020703 permutation N at -X,Y
301 step=2, n_start=0
302 A005563 N on X=Y diagonal, Y*(Y+2)
303 A000290 N on X=-Y North-West diagonal (start row), Y^2
304 step=2, n_start=2
305 A059100 N on north-west diagonal (start each row), Y^2+2
306 A053615 abs(X), runs k..0..k
307 step=2, align=right, n_start=0
308 A196199 X-Y, runs -k to +k
309 A053615 abs(X-Y), runs k..0..k
310 step=2, align=left, n_start=0
311 A005563 N on Y axis, Y*(Y+2)
312 step=3
314 A180447 Y coordinate, n appears 3n+1 times
315 A104249 N on Y axis, Y*(3Y+1)/2+1
316 A143689 N on X=-Y North-West diagonal
317 step=3, n_start=0
318 A005449 N on Y axis, second pentagonals Y*(3Y+1)/2
319 A000326 N on diagonal north-west, pentagonals Y*(3Y-1)/2
320 step=4
322 A084849 N on Y axis
323 A001844 N on X=Y diagonal (North-East)
324 A058331 N on X=-Y North-West diagonal
325 A221217 permutation N at -X,Y
326 step=4, n_start=0
327 A014105 N on Y axis, the second hexagonal numbers
328 A046092 N on X=Y diagonal, 4*triangular numbers
329 step=4, align=right, n_start=0
330 A060511 X coordinate, amount n exceeds hexagonal number
331 A000384 N on Y axis, the hexagonal numbers
332 A001105 N on X=Y diagonal, 2*squares
333 step=5
335 A116668 N on Y axis
336 step=6
338 A056108 N on Y axis
339 A056109 N on X=Y diagonal (North-East)
340 A056107 N on X=-Y North-West diagonal
341 step=8
343 A053755 N on X=-Y North-West diagonal
344 step=9
346 A006137 N on Y axis
347 A038764 N on X=Y diagonal (North-East)
348
350 Math::PlanePath, Math::PlanePath::PyramidSides,
351 Math::PlanePath::Corner, Math::PlanePath::SacksSpiral,
352 Math::PlanePath::MultipleRings
353 Math::PlanePath::Diagonals, Math::PlanePath::DiagonalsOctant,
355 Math::PlanePath::Rows
356
358 <http://user42.tuxfamily.org/math-planepath/index.html>
359
361 Copyright 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
362 2020 Kevin Ryde
363 This file is part of Math-PlanePath.
365 Math-PlanePath is free software; you can redistribute it and/or modify
367 it under the terms of the GNU General Public License as published by
368 the Free Software Foundation; either version 3, or (at your option) any
369 later version.
370 Math-PlanePath is distributed in the hope that it will be useful, but
372 WITHOUT ANY WARRANTY; without even the implied warranty of
373 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
374 General Public License for more details.
375 You should have received a copy of the GNU General Public License along
377 with Math-PlanePath. If not, see <http://www.gnu.org/licenses/>.
378perl v5.34.0 2021-07-22 Math::PlanePath::PyramidRows(3)