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(Centered polygons) pyramidal numbers

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The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers are a family of sequences of 3-dimensional nonregular polytope numbers (among the 3-dimensional figurate numbers) formed by adding the first [N0-1] positive centered polygonal numbers with constant number of sides [N0-1], where N0 is the number of vertices (including the apex vertex) of the polygonal base pyramid. The term centered pyramid numbers, i.e. (centered squares) pyramidal numbers, is often used to refer to the centered square pyramidal numbers, i.e. (centered squares) pyramidal numbers, having a polygonal base with four sides. The centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, are a generalization of the centered pyramid numbers, i.e. (centered squares) pyramidal numbers, where the base is a regular convex polygon with any number of sides [N0-1] ≥ 3. Centered pyramidal numbers, i.e. (centered polygons) pyramidal numbers, may also be generalized to higher dimensions as centered hyperpyramidal numbers, i.e. (centered polygons) hyperpyramidal numbers.

While the (centered polygons) pyramidal numbers are pyramidal stacks of centered polygons, the generated figures are NOT GLOBALLY CENTERED. Thus the (centered polygons) pyramidal numbers do not belong to the category of globally centered figurate numbers (which start with the globally central dot, giving value 1, for n = 0,) they belong to the category of globally noncentered figurate numbers (which equals 0 for n = 0 and start with the initial dot, giving value 1, for n = 1.) It would be less confusing if the centered pyramidal numbers were called (centered polygons) pyramidal numbers. Note that although the triangular pyramidal numbers are tetrahedral numbers, the (centered triangles) pyramidal numbers are NOT the (globally centered) centered tetrahedral numbers.


All figurate numbers are accessible via this structured menu: Classifications of figurate numbers

Contents

Formulae

The nth [N0-1]-gonal base centered pyramidal number is given by the formula: [1]

\,_{'c'}Y^{(3)}_{N_0}(n) \equiv \sum_{i = 0}^n \,_cP^{(2)}_{[N_0-1]}(i) = \frac{n\{[N_0-1] n^2 - ([N_0-1] - 6)\}}{6} = [N_0-1] \frac{(n-1)n(n+1)}{6} + n = [N_0-1] \binom{n+1}{3} + n,\,

where _cP^{(2)}_{N_0}(n)\, is the nth N0-sided centered polygonal number.

Observe that for [N0-1] = 6, the formula simplifies to that of the cubes which means that the nth centered hexagonal pyramidal number may be rearranged as the nth cube!

Descartes-Euler (convex) polyhedral formula

Descartes-Euler (convex) polyhedral formula:[2]

{\sum_{i=0}^2 (-1)^i N_i} = N_0-N_1+N_2 = V-E+F = 2,\,

where N0 is the number of 0-dimensional elements (vertices V,) N1 is the number of 1-dimensional elements (edges E) and N2 is the number of 2-dimensional elements (faces F) of the polyhedron.

Recurrence relation

\,_{'c'}Y^{(3)}_{N_0}(n) = ?\,

with initial conditions

\,_{'c'}Y^{(3)}_{N_0}(0) = ?\,

Generating function

G_{\{\,_{'c'}Y^{(3)}_{N_0}(n)\}}(x) = \frac{x(1+([N_0-1]-2)x+x^2)}{(x-1)^4} = \sum_{n=0}^\infty \frac{n (7-n^2+(-1+n^2) N_0)}{6} x^n\,

Order of basis

In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and k k-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat's proof has never been found.[3] Joseph Louis Lagrange proved the square case (known as the four squares theorem) in 1770 and Gauss proved the triangular case in 1796. In 1813, Cauchy finally proved the horizontal generalization that every nonnegative integer can be written as a sum of k k-gon numbers (known as the polygonal number theorem,) while a vertical (higher dimensional) generalization has also been made (known as the Hilbert-Waring problem.)

A nonempty subset \scriptstyle A\, of nonnegative integers is called a basis of order \scriptstyle g\, if \scriptstyle g\, is the minimum number with the property that every nonnegative integer can be written as a sum of \scriptstyle g\, elements in \scriptstyle A\,. Lagrange’s sum of four squares can be restated as the set \scriptstyle \{n^2|n = 0, 1, 2, \ldots\}\, of nonnegative squares forms a basis of order 4.

Theorem (Cauchy) For every \scriptstyle k \ge 3, the set \scriptstyle \{P(k, n)|n = 0, 1, 2, \ldots\}\, of k-gon numbers forms a basis of order \scriptstyle k\,, i.e. every nonnegative integer can be written as a sum of \scriptstyle k\, k-gon numbers.

We note that polygonal numbers are two dimensional analogues of squares. Obviously, cubes, fourth powers, fifth powers, ... are higher dimensional analogues of squares. In 1770, Waring stated without proof that every nonnegative integer can be written as a sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 1909, Hilbert proved that there is a finite number \scriptstyle g(d)\, such that every nonnegative integer is a sum of \scriptstyle g(d)\, \scriptstyle d\,th powers, i.e. the set \scriptstyle \{n^d|n = 0, 1, 2, \ldots\}\, of \scriptstyle d\,th powers forms a basis of order \scriptstyle g(d)\,. The Hilbert-Waring problem is concerned with the study of \scriptstyle g(d)\, for \scriptstyle d \ge 2\,. This problem was one of the most important research topics in additive number theory in last 90 years, and it is still a very active area of research.

Differences

\,_{'c'}Y^{(3)}_{N_0}(n) - \,_{'c'}Y^{(3)}_{N_0}(n-1) = \,_cP^{(2)}_{[N_0-1]}(n)\,

Partial sums

\sum_{n=1}^m {\,_{'c'}Y^{(3)}_{N_0}(n)} = ?\,

Partial sums of reciprocals

\sum_{n=1}^m \frac{1}{\,_{'c'}Y^{(3)}_{N_0}(n)} = ?\,

Sum of reciprocals

\sum_{n=1}^{\infty} \frac{1}{\,_{'c'}Y^{(3)}_{N_0}(n)} = ?\,

Table of formulae and values

(Centered polygons) pyramidal numbers formulae and values
N0−1 Name Formulae

\,_{'c'}Y^{(3)}_{N_0}(n) =\,


\scriptstyle \frac{n\{[N_0-1] n^2 - ([N_0-1] - 6)\}}{6}\,

n = 0 1 2 3 4 5 6 7 8 9 10 11 12 OEIS

number

3 Centered triangular pyramidal \frac{n(3n^2 + 3)}{6}\,


\frac{n(n^2 + 1)}{2}\,

0 1 5 15 34 65 111 175 260 369 505 671 870 A006003
4 Centered square pyramidal \frac{n(4n^2 + 2)}{6}\,


\frac{n(2n^2 + 1)}{3}\,


P^{(3)}_{2\cdot3}(n)\,


Octahedral

0 1 6 19 44 85 146 231 344 489 670 891 1156 A005900
5 Centered pentagonal pyramidal \frac{n(5n^2 + 1)}{6}\, 0 1 7 23 54 105 181 287 428 609 835 1111 1442 A004068
6 Centered hexagonal pyramidal \frac{n(6n^2 + 0)}{6}\,


n^3\,


P^{(3)}_{2^3}(n)\,


Cubic

0 1 8 27 64 125 216 343 512 729 1000 1331 1728 A000578
7 Centered heptagonal pyramidal \frac{n(7n^2 - 1)}{6}\, 0 1 9 31 74 145 251 399 596 849 1165 1551 2014 A004126
8 Centered octagonal pyramidal \frac{n(8n^2 - 2)}{6}\,


\frac{n(4n^2 - 1)}{3}\,

0 1 10 35 84 165 286 455 680 969 1330 1771 2300 A000447
9 Centered nonagonal pyramidal \frac{n(9n^2-3)}{6}\,


\frac{n(3n^2-1)}{2}\,

0 1 11 39 94 185 321 511 764 1089 1495 1991 2586 A004188
10 Centered decagonal pyramidal \frac{n(10n^2-4)}{6}\,


\frac{n(5n^2-2)}{3}\,

0 1 12 43 104 205 356 567 848 1209 1660 2211 2872 A004466
11 Centered hendecagonal pyramidal \frac{n(11n^2-5)}{6}\, 0 1 13 47 114 225 391 623 932 1329 1825 2431 3158 A004467
12 Centered dodecagonal pyramidal \frac{n(12n^2-6)}{6}\,


n(2n^2 - 1)\,


\scriptstyle P^{(3)}_{2\cdot3}(n) + 8 P^{(3)}_{3+1}(n-1)\,


Stellated octahedron [4]


(Stella octangula) [5]

0 1 14 51 124 245 426 679 1016 1449 1990 2651 3444 A007588
13 Centered tridecagonal pyramidal \frac{n(13n^2-7)}{6}\, 0 1 15 55 134 265 461 735 1100 1569 2155 2871 3730 A062025
14 Centered tetradecagonal pyramidal \frac{n(14n^2-8)}{6}\,


\frac{n(7n^2-4)}{3}\,

0 1 16 59 144 285 496 791 1184 1689 2320 3091 4016 A063521
15 Centered pentadecagonal pyramidal \frac{n(15n^2-9)}{6}\,


\frac{n(5n^2-3)}{2}\,

0 1 17 63 154 305 531 847 1268 1809 2485 3311 4302 A063522
16 Centered hexadecagonal pyramidal \frac{n(16n^2-10)}{6}\,


\frac{n(8n^2-5)}{3}\,

0 1 18 67 164 325 566 903 1352 1929 2650 3531 4588 A063523
17 Centered heptadecagonal pyramidal \frac{n(17n^2-11)}{6}\, 0 1 19                     A??????
18 Centered octadecagonal pyramidal \frac{n(18n^2-12)}{6}\, 0 1 20                     A??????
19 Centered nonadecagonal pyramidal \frac{n(19n^2-13)}{6}\, 0 1 21                     A??????
20 Centered icosagonal pyramidal \frac{n(20n^2-14)}{6}\, 0 1 22                     A??????
21 Centered icosihenagonal pyramidal \frac{n(21n^2-15)}{6}\, 0 1 23                     A??????
22 Centered icosidigonal pyramidal \frac{n(22n^2-16)}{6}\, 0 1 24                     A??????
23 Centered icositrigonal pyramidal \frac{n(23n^2-17)}{6}\, 0 1 25                     A??????
24 Centered icositetragonal pyramidal \frac{n(24n^2-18)}{6}\, 0 1 26                     A??????
25 Centered icosipentagonal pyramidal \frac{n(25n^2-19)}{6}\, 0 1 27                     A??????
26 Centered icosihexagonal pyramidal \frac{n(26n^2-20)}{6}\, 0 1 28                     A??????
27 Centered icosiheptagonal pyramidal \frac{n(27n^2-21)}{6}\, 0 1 29                     A??????
28 Centered icosioctagonal pyramidal \frac{n(28n^2-22)}{6}\, 0 1 30                     A??????
29 Centered icosinonagonal pyramidal \frac{n(29n^2-23)}{6}\, 0 1 31                     A??????
30 Centered triacontagonal pyramidal \frac{n(30n^2-24)}{6}\, 0 1 32                     A??????


Table of related formulae and values

(Centered polygons) pyramidal numbers related formulae and values
N0−1 Generating

function

G_{\{\,_{'c'}Y^{(3)}_{N_0}(n)\}}(x) =\,


\scriptstyle \frac{x(1+([N_0-1]-2)x+x^2)}{(x-1)^4}\,

Order

of basis

g_{\{\,_{'c'}Y^{(3)}_{N_0}\}} =\,

Differences

\,_{'c'}Y^{(3)}_{N_0}(n) - \,

\,_{'c'}Y^{(3)}_{N_0}(n-1) =\,


\,_cP^{(2)}_{[N_0-1]}(n)\,

Partial sums

\sum_{n=1}^m {\,_{'c'}Y^{(3)}_{N_0}(n)} =\,

Partial sums of reciprocals

\sum_{n=1}^m {1\over{\,_{'c'}Y^{(3)}_{N_0}(n)}} =\,

Sum of Reciprocals[6][7]

\sum_{n=1}^\infty{1\over{\,_{'c'}Y^{(3)}_{N_0}(n)}} =\,

3 \frac{x(1+x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{3}(n)\, \, \,
4 \frac{x(1+2x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{4}(n)\, \, \,
5 \frac{x(1+3x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{5}(n)\, \, \,
6 \frac{x(1+4x+x^2)}{(1-x)^4}\, \, \,_cP^{(2)}_{6}(n)\, \, \,
7 \frac{x(1+5x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{7}(n)\, \, \,
8 \frac{x(1+6x+x^2)}{(1-x)^4}\, \, \,_cP^{(2)}_{8}(n)\, \, \,
9 \frac{x(1+7x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{9}(n)\, \, \,
10 \frac{x(1+8x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{10}(n)\, \, \,
11 \frac{x(1+9x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{11}(n)\, \, \,
12 \frac{x(1+10x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{12}(n)\, \, \,
13 \frac{x(1+11x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{13}(n)\, \, \,
14 \frac{x(1+12x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{14}(n)\, \, \,
15 \frac{x(1+13x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{15}(n)\, \, \,
16 \frac{x(1+14x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{16}(n)\, \, \,
17 \frac{x(1+15x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{17}(n)\, \, \,
18 \frac{x(1+16x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{18}(n)\, \, \,
19 \frac{x(1+17x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{19}(n)\, \, \,
20 \frac{x(1+18x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{20}(n)\, \, \,
21 \frac{x(1+19x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{21}(n)\, \, \,
22 \frac{x(1+20x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{22}(n)\, \, \,
23 \frac{x(1+21x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{23}(n)\, \, \,
24 \frac{x(1+22x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{24}(n)\, \, \,
25 \frac{x(1+23x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{25}(n)\, \, \,
26 \frac{x(1+24x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{26}(n)\, \, \,
27 \frac{x(1+25x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{27}(n)\, \, \,
28 \frac{x(1+26x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{28}(n)\, \, \,
29 \frac{x(1+27x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{29}(n)\, \, \,
30 \frac{x(1+28x+x^2)}{(x-1)^4}\, \, \,_cP^{(2)}_{30}(n)\, \, \,


Table of sequences

(Centered polygons) pyramidal numbers sequences
N0−1 \,_{'c'}Y^{(3)}_{N_0}(n),\ n \ge 0\, sequences
3 {1, 5, 15, 34, 65, 111, 175, 260, 369, 505, 671, 870, 1105, 1379, 1695, 2056, 2465, 2925, 3439, 4010, 4641, 5335, 6095, 6924, 7825, 8801, 9855, 10990, 12209, 13515, 14911, ...}
4 {1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891, 1156, 1469, 1834, 2255, 2736, 3281, 3894, 4579, 5340, 6181, 7106, 8119, 9224, 10425, 11726, 13131, 14644, 16269, 18010, ...}
5 {1, 7, 23, 54, 105, 181, 287, 428, 609, 835, 1111, 1442, 1833, 2289, 2815, 3416, 4097, 4863, 5719, 6670, 7721, 8877, 10143, 11524, 13025, 14651, 16407, 18298, 20329, 22505, ...}
6 {1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728, 2197, 2744, 3375, 4096, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 15625, 17576, 19683, 21952, 24389, ...}
7 {1, 9, 31, 74, 145, 251, 399, 596, 849, 1165, 1551, 2014, 2561, 3199, 3935, 4776, 5729, 6801, 7999, 9330, 10801, 12419, 14191, 16124, 18225, 20501, 22959, 25606, 28449, ...}
8 {1, 10, 35, 84, 165, 286, 455, 680, 969, 1330, 1771, 2300, 2925, 3654, 4495, 5456, 6545, 7770, 9139, 10660, 12341, 14190, 16215, 18424, 20825, 23426, 26235, 29260, 32509, ...}
9 {1, 11, 39, 94, 185, 321, 511, 764, 1089, 1495, 1991, 2586, 3289, 4109, 5055, 6136, 7361, 8739, 10279, 11990, 13881, 15961, 18239, 20724, 23425, 26351, 29511, 32914, ...}
10 {1, 12, 43, 104, 205, 356, 567, 848, 1209, 1660, 2211, 2872, 3653, 4564, 5615, 6816, 8177, 9708, 11419, 13320, 15421, 17732, 20263, 23024, 26025, 29276, 32787, 36568, ...}
11 {1, 13, 47, 114, 225, 391, 623, 932, 1329, 1825, 2431, 3158, 4017, 5019, 6175, 7496, 8993, 10677, 12559, 14650, 16961, 19503, 22287, 25324, 28625, 32201, 36063, 40222, ...}
12 {1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, 5474, 6735, 8176, 9809, 11646, 13699, 15980, 18501, 21274, 24311, 27624, 31225, 35126, 39339, 43876, ...}
13 {1, 15, 55, 134, 265, 461, 735, 1100, 1569, 2155, 2871, 3730, 4745, 5929, 7295, 8856, 10625, 12615, 14839, 17310, 20041, 23045, 26335, 29924, 33825, 38051, 42615, 47530, ...}
14 {1, 16, 59, 144, 285, 496, 791, 1184, 1689, 2320, 3091, 4016, 5109, 6384, 7855, 9536, 11441, 13584, 15979, 18640, 21581, 24816, 28359, 32224, 36425, 40976, 45891, 51184, ...}
15 {1, 17, 63, 154, 305, 531, 847, 1268, 1809, 2485, 3311, 4302, 5473, 6839, 8415, 10216, 12257, 14553, 17119, 19970, 23121, 26587, 30383, 34524, 39025, 43901, 49167, 54838, ...}
16 {1, 18, 67, 164, 325, 566, 903, 1352, 1929, 2650, 3531, 4588, 5837, 7294, 8975, 10896, 13073, 15522, 18259, 21300, 24661, 28358, 32407, 36824, 41625, 46826, 52443, ...}
17 {1, 19, 71, 174, 345, 601, 959, 1436, 2049, 2815, 3751, 4874, 6201, 7749, 9535, 11576, 13889, 16491, 19399, 22630, 26201, 30129, 34431, 39124, 44225, 49751, 55719, ...}
18 {1, 20, 75, 184, 365, 636, 1015, 1520, 2169, 2980, 3971, 5160, 6565, 8204, 10095, 12256, 14705, 17460, 20539, 23960, 27741, 31900, 36455, 41424, 46825, 52676, 58995, ...}
19 {1, 21, 79, 194, 385, 671, 1071, 1604, 2289, 3145, 4191, 5446, 6929, 8659, 10655, 12936, 15521, 18429, 21679, 25290, 29281, 33671, 38479, 43724, 49425, 55601, 62271, ...}
20 {1, 22, 83, 204, 405, 706, 1127, 1688, 2409, 3310, 4411, 5732, 7293, 9114, 11215, 13616, 16337, 19398, 22819, 26620, 30821, 35442, 40503, 46024, 52025, 58526, 65547, ...}
21 {1, 23, 87, 214, 425, 741, 1183, 1772, 2529, 3475, 4631, 6018, 7657, 9569, 11775, 14296, 17153, 20367, 23959, 27950, 32361, 37213, 42527, 48324, 54625, 61451, 68823, ...}
22 {1, 24, 91, 224, 445, 776, 1239, 1856, 2649, 3640, 4851, 6304, 8021, 10024, 12335, 14976, 17969, 21336, 25099, 29280, 33901, 38984, 44551, 50624, 57225, 64376, 72099, ...}
23 {1, 25, 95, 234, 465, 811, 1295, 1940, 2769, 3805, 5071, 6590, 8385, 10479, 12895, 15656, 18785, 22305, 26239, 30610, 35441, 40755, 46575, 52924, 59825, 67301, 75375, ...}
24 {1, 26, 99, 244, 485, 846, 1351, 2024, 2889, 3970, 5291, 6876, 8749, 10934, 13455, 16336, 19601, 23274, 27379, 31940, 36981, 42526, 48599, 55224, 62425, 70226, 78651, ...}
25 {1, 27, 103, 254, 505, 881, 1407, 2108, 3009, 4135, 5511, 7162, 9113, 11389, 14015, 17016, 20417, 24243, 28519, 33270, 38521, 44297, 50623, 57524, 65025, 73151, 81927, ...}
26 {1, 28, 107, 264, 525, 916, 1463, 2192, 3129, 4300, 5731, 7448, 9477, 11844, 14575, 17696, 21233, 25212, 29659, 34600, 40061, 46068, 52647, 59824, 67625, 76076, 85203, ...}
27 {1, 29, 111, 274, 545, 951, 1519, 2276, 3249, 4465, 5951, 7734, 9841, 12299, 15135, 18376, 22049, 26181, 30799, 35930, 41601, 47839, 54671, 62124, 70225, 79001, 88479, ...}
28 {1, 30, 115, 284, 565, 986, 1575, 2360, 3369, 4630, 6171, 8020, 10205, 12754, 15695, 19056, 22865, 27150, 31939, 37260, 43141, 49610, 56695, 64424, 72825, 81926, 91755, ...}
29 {1, 31, 119, 294, 585, 1021, 1631, 2444, 3489, 4795, 6391, 8306, 10569, 13209, 16255, 19736, 23681, 28119, 33079, 38590, 44681, 51381, 58719, 66724, 75425, 84851, 95031, ...}
30 {1, 32, 123, 304, 605, 1056, 1687, 2528, 3609, 4960, 6611, 8592, 10933, 13664, 16815, 20416, 24497, 29088, 34219, 39920, 46221, 53152, 60743, 69024, 78025, 87776, 98307, ...}


See also

Pyramidal numbers

Notes

  1. Where \scriptstyle {\,}_{'c'}Y^{(d)}_{[(k+2)+(d-2)]}(n) = {\,}_{'c'}Y^{(d)}_{k+d}(n)\,, k ≥ 1, n ≥ 0, is the d-dimensional, d ≥ 0, (k+2)-gonal base (centered polygons) (hyper)pyramidal number where, for d ≥ 2, \scriptstyle N_0 = [(k+2)+(d-2)]\, is the number of vertices (including the \scriptstyle d-2\, apex vertices) of the (centered polygonal base) (hyper)pyramid (the quoted \scriptstyle 'c'\, emphasizes that only the polygons are centered, not the whole figure.)
  2. Weisstein, Eric W., Polyhedral Formula, From MathWorld--A Wolfram Web Resource.
  3. Weisstein, Eric W., Fermat's Polygonal Number Theorem, From MathWorld--A Wolfram Web Resource.
  4. Weisstein, Eric W., Stellated octahedron, From MathWorld--A Wolfram Web Resource.
  5. Weisstein, Eric W., Stella Octangula Number, From MathWorld--A Wolfram Web Resource.
  6. Downey, Lawrence M., Ong, Boon W., and Sellers, James A., Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, 2008.
  7. PSYCHEDELIC GEOMETRY, INVERSE POLYGONAL NUMBERS SERIES.

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