1math::figurate(n) Tcl Math Library math::figurate(n)
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6
8 math::figurate - Evaluate figurate numbers
9
11 package require Tcl 8.6
12 package require math::figurate 1.0
14 ::math::figurate::sum_sequence n
16 ::math::figurate::sum_squares n
18 ::math::figurate::sum_cubes n
20 ::math::figurate::sum_4th_power n
22 ::math::figurate::sum_5th_power n
24 ::math::figurate::sum_6th_power n
26 ::math::figurate::sum_7th_power n
28 ::math::figurate::sum_8th_power n
30 ::math::figurate::sum_9th_power n
32 ::math::figurate::sum_10th_power n
34 ::math::figurate::sum_sequence_odd n
36 ::math::figurate::sum_squares_odd n
38 ::math::figurate::sum_cubes_odd n
40 ::math::figurate::sum_4th_power_odd n
42 ::math::figurate::sum_5th_power_odd n
44 ::math::figurate::sum_6th_power_odd n
46 ::math::figurate::sum_7th_power_odd n
48 ::math::figurate::sum_8th_power_odd n
50 ::math::figurate::sum_9th_power_odd n
52 ::math::figurate::sum_10th_power_odd n
54 ::math::figurate::oblong n
56 ::math::figurate::pronic n
58 ::math::figurate::triangular n
60 ::math::figurate::square n
62 ::math::figurate::cubic n
64 ::math::figurate::biquadratic n
66 ::math::figurate::centeredTriangular n
68 ::math::figurate::centeredSquare n
70 ::math::figurate::centeredPentagonal n
72 ::math::figurate::centeredHexagonal n
74 ::math::figurate::centeredCube n
76 ::math::figurate::decagonal n
78 ::math::figurate::heptagonal n
80 ::math::figurate::hexagonal n
82 ::math::figurate::octagonal n
84 ::math::figurate::octahedral n
86 ::math::figurate::pentagonal n
88 ::math::figurate::squarePyramidral n
90 ::math::figurate::tetrahedral n
92 ::math::figurate::pentatope n
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96
98 Sums of numbers that follow a particular pattern are called figurate
99 numbers. A simple example is the sum of integers 1, 2, ... up to n.
100 You can arrange 1, 1+2=3, 1+2+3=6, ... objects in a triangle, hence the
101 name triangular numbers:
102 *
105 * *
106 * * *
107 * * * *
108 ...
109 The math::figurate package consists of a collection of procedures to
111 evaluate a wide variety of figurate numbers. While all formulae are
112 straightforward, the details are sometimes puzzling. Note: The proce‐
113 dures consider arguments lower than zero as to mean "no objects to be
114 counted" and therefore return 0.
115
117 The procedures can be arranged in a few categories: sums of integers
118 raised to a particular power, sums of odd integers and general figurate
119 numbers, for instance the pentagonal numbers.
120 ::math::figurate::sum_sequence n
122 Return the sum of integers 1, 2, ..., n.
123 int n Highest integer in the sum
125 ::math::figurate::sum_squares n
128 Return the sum of squares 1**2, 2**2, ..., n**2.
129 int n Highest base integer in the sum
131 ::math::figurate::sum_cubes n
134 Return the sum of cubes 1**3, 2**3, ..., n**3.
135 int n Highest base integer in the sum
137 ::math::figurate::sum_4th_power n
140 Return the sum of 4th powers 1**4, 2**4, ..., n**4.
141 int n Highest base integer in the sum
143 ::math::figurate::sum_5th_power n
146 Return the sum of 5th powers 1**5, 2**5, ..., n**5.
147 int n Highest base integer in the sum
149 ::math::figurate::sum_6th_power n
152 Return the sum of 6th powers 1**6, 2**6, ..., n**6.
153 int n Highest base integer in the sum
155 ::math::figurate::sum_7th_power n
158 Return the sum of 7th powers 1**7, 2**7, ..., n**7.
159 int n Highest base integer in the sum
161 ::math::figurate::sum_8th_power n
163 Return the sum of 8th powers 1**8, 2**8, ..., n**8.
164 int n Highest base integer in the sum
166 ::math::figurate::sum_9th_power n
169 Return the sum of 9th powers 1**9, 2**9, ..., n**9.
170 int n Highest base integer in the sum
172 ::math::figurate::sum_10th_power n
175 Return the sum of 10th powers 1**10, 2**10, ..., n**10.
176 int n Highest base integer in the sum
178 ::math::figurate::sum_sequence_odd n
181 Return the sum of odd integers 1, 3, ..., 2n-1
182 int n Highest integer in the sum
184 ::math::figurate::sum_squares_odd n
187 Return the sum of odd squares 1**2, 3**2, ..., (2n-1)**2.
188 int n Highest base integer in the sum
190 ::math::figurate::sum_cubes_odd n
193 Return the sum of odd cubes 1**3, 3**3, ..., (2n-1)**3.
194 int n Highest base integer in the sum
196 ::math::figurate::sum_4th_power_odd n
199 Return the sum of odd 4th powers 1**4, 2**4, ..., (2n-1)**4.
200 int n Highest base integer in the sum
202 ::math::figurate::sum_5th_power_odd n
205 Return the sum of odd 5th powers 1**5, 2**5, ..., (2n-1)**5.
206 int n Highest base integer in the sum
208 ::math::figurate::sum_6th_power_odd n
211 Return the sum of odd 6th powers 1**6, 2**6, ..., (2n-1)**6.
212 int n Highest base integer in the sum
214 ::math::figurate::sum_7th_power_odd n
217 Return the sum of odd 7th powers 1**7, 2**7, ..., (2n-1)**7.
218 int n Highest base integer in the sum
220 ::math::figurate::sum_8th_power_odd n
222 Return the sum of odd 8th powers 1**8, 2**8, ..., (2n-1)**8.
223 int n Highest base integer in the sum
225 ::math::figurate::sum_9th_power_odd n
228 Return the sum of odd 9th powers 1**9, 2**9, ..., (2n-1)**9.
229 int n Highest base integer in the sum
231 ::math::figurate::sum_10th_power_odd n
234 Return the sum of odd 10th powers 1**10, 2**10, ..., (2n-1)**10.
235 int n Highest base integer in the sum
237 ::math::figurate::oblong n
240 Return the nth oblong number (twice the nth triangular number)
241 int n Required index
243 ::math::figurate::pronic n
246 Return the nth pronic number (synonym for oblong)
247 int n Required index
249 ::math::figurate::triangular n
252 Return the nth triangular number
253 int n Required index
255 ::math::figurate::square n
258 Return the nth square number
259 int n Required index
261 ::math::figurate::cubic n
264 Return the nth cubic number
265 int n Required index
267 ::math::figurate::biquadratic n
270 Return the nth biquaratic number (i.e. n**4)
271 int n Required index
273 ::math::figurate::centeredTriangular n
276 Return the nth centered triangular number (items arranged in
277 concentric squares)
278 int n Required index
280 ::math::figurate::centeredSquare n
283 Return the nth centered square number (items arranged in concen‐
284 tric squares)
285 int n Required index
287 ::math::figurate::centeredPentagonal n
290 Return the nth centered pentagonal number (items arranged in
291 concentric pentagons)
292 int n Required index
294 ::math::figurate::centeredHexagonal n
297 Return the nth centered hexagonal number (items arranged in con‐
298 centric hexagons)
299 int n Required index
301 ::math::figurate::centeredCube n
304 Return the nth centered cube number (items arranged in concen‐
305 tric cubes)
306 int n Required index
308 ::math::figurate::decagonal n
311 Return the nth decagonal number (items arranged in decagons with
312 one common vertex)
313 int n Required index
315 ::math::figurate::heptagonal n
318 Return the nth heptagonal number (items arranged in heptagons
319 with one common vertex)
320 int n Required index
322 ::math::figurate::hexagonal n
325 Return the nth hexagonal number (items arranged in hexagons with
326 one common vertex)
327 int n Required index
329 ::math::figurate::octagonal n
332 Return the nth octagonal number (items arranged in octagons with
333 one common vertex)
334 int n Required index
336 ::math::figurate::octahedral n
339 Return the nth octahedral number (items arranged in octahedrons
340 with a common centre)
341 int n Required index
343 ::math::figurate::pentagonal n
346 Return the nth pentagonal number (items arranged in pentagons
347 with one common vertex)
348 int n Required index
350 ::math::figurate::squarePyramidral n
353 Return the nth square pyramidral number (items arranged in a
354 square pyramid)
355 int n Required index
357 ::math::figurate::tetrahedral n
360 Return the nth tetrahedral number (items arranged in a triangu‐
361 lar pyramid)
362 int n Required index
364 ::math::figurate::pentatope n
367 Return the nth pentatope number (items arranged in the four-di‐
368 mensional analogue of a triangular pyramid)
369 int n Required index
371
374 This document, and the package it describes, will undoubtedly contain
375 bugs and other problems. Please report such in the category math ::
376 figurate of the Tcllib Trackers [http://core.tcl.tk/tcllib/reportlist].
377 Please also report any ideas for enhancements you may have for either
378 package and/or documentation.
379 When proposing code changes, please provide unified diffs, i.e the out‐
381 put of diff -u.
382 Note further that attachments are strongly preferred over inlined
384 patches. Attachments can be made by going to the Edit form of the
385 ticket immediately after its creation, and then using the left-most
386 button in the secondary navigation bar.
387
389 figurate numbers, mathematics
390
392 Mathematics
393tcllib 1.0 math::figurate(n)