Alejandro J. Becerra Jr. - cp's OEIS frontend

A318159 Figurate numbers based on the small stellated dodecahedron: a(n) = n(21n^2 - 33*n + 14)/2.

1, 32, 156, 436, 935, 1716, 2842, 4376, 6381, 8920, 12056, 15852, 20371, 25676, 31830, 38896, 46937, 56016, 66196, 77540, 90111, 103972, 119186, 135816, 153925, 173576, 194832, 217756, 242411, 268860, 297166, 327392, 359601, 393856, 430220, 468756, 509527

Offset: 1

Alejandro J. Becerra Jr., Aug 19 2018

The small stellated dodecahedron is a 3D nonconvex regular polyhedron represented by the Schlaefli symbol {5/2, 5}.
When truncated, a degenerate dodecahedron is produced.  It is then easy to recognize that every small stellated dodecahedron can be constructed by morphing the 12 pentagonal faces of a regular dodecahedron into pentagonal pyramids.
The last digits form a cycle of length 20 [1, 2, 6, 6, ..., 1, 2, 6, 6].

Colin Barker, Table of n, a(n) for n = 1..1000
Wikipedia, Small stellated dodecahedron.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A002411, A006566.

[n*(21*n^2-33*n+14)/2: n in [1..40]]; // Vincenzo Librandi, Aug 27 2018

Table[(n (14 - 33 n + 21 n^2)) / 2, {n, 45}] (* Vincenzo Librandi, Aug 27 2018 *)
CoefficientList[Series[(1 + 28*x + 34*x^2) / (1 - x)^4 , {x, 0, 45}], x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 32, 156, 436}, 45] (* Stefano Spezia, Sep 02 2018 *)

Vec(x*(1 + 28*x + 34*x^2) / (1 - x)^4 + O(x^40)) \\ Colin Barker, Aug 20 2018

a(n) = (n*(14 - 33*n + 21*n^2)) / 2 \\ Colin Barker, Aug 20 2018

a(n) = A006566(n) + 12*A002411(n-1).
a(n) == a(n+20) (mod 10).
From Colin Barker, Aug 20 2018: (Start)
G.f.: x*(1 + 28*x + 34*x^2)/(1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4. (End)
E.g.f.: exp(x)*x*(2 + 30*x + 21*x^2)/2. - Elmo R. Oliveira, Aug 22 2025

More terms from Colin Barker, Aug 20 2018

A300624 Figurate numbers based on the 11-dimensional regular convex polytope called the 11-dimensional cross-polytope, or 11-dimensional hyperoctahedron.

Original entry on oeis.org

0, 1, 22, 243, 1804, 10165, 46530, 180775, 614680, 1871145, 5188590, 13286043, 31760676, 71513949, 152784282, 311603535, 609802800, 1150082385, 2098144710, 3714481475, 6399123260, 10753517061, 17664712562, 28418229623, 44847366984, 69528316025, 106032285086

Offset: 0

Alejandro J. Becerra Jr., Aug 14 2018

The 11-dimensional cross-polytope is represented by the Schlaefli symbol {3, 3, 3, 3, 3, 3, 3, 3, 3, 4}. It is the dual of the 11-dimensional hypercube.

Georg Fischer, Table of n, a(n) for n = 0..100 (first 61 terms from Alejandro J. Becerra Jr.)
Index entries for linear recurrences with constant coefficients, signature (12,-66,220,-495,792,-924,792,-495,220,-66,12,-1).

Similar sequences: A005900 (m=3), A014820(n-1) (m=4), A069038 (m=5), A069039 (m=6), A099193 (m=7), A099195 (m=8), A099196 (m=9), A099197 (m=10).

[(n*(14175 + 83754*n^2 + 50270*n^4 + 7392*n^6 + 330*n^8 + 4*n^10)) / 155925 : n in [0..40]]; // Wesley Ivan Hurt, Jul 17 2020

concat(0, Vec(x*(1 + x)^10 / (1 - x)^12 + O(x^40))) \\ Colin Barker, Aug 15 2018

a(n) = (n*(14175 + 83754*n^2 + 50270*n^4 + 7392*n^6 + 330*n^8 + 4*n^10)) / 155925 \\ Colin Barker, Aug 15 2018

a(n) = 11-crosspolytope(n).
From Colin Barker, Aug 15 2018: (Start)
G.f.: x*(1 + x)^10 / (1 - x)^12.
a(n) = (n*(14175 + 83754*n^2 + 50270*n^4 + 7392*n^6 + 330*n^8 + 4*n^10)) / 155925.
(End)

A302562 Partial sums of A092181.

Original entry on oeis.org

1, 25, 178, 722, 2147, 5243, 11172, 21540, 38469, 64669, 103510, 159094, 236327, 340991, 479816, 660552, 892041, 1184289, 1548538, 1997338, 2544619, 3205763, 3997676, 4938860, 6049485, 7351461, 8868510, 10626238, 12652207, 14976007, 17629328, 20646032

Offset: 1

Alejandro J. Becerra Jr., Aug 15 2018

Geometrically, the partial sums of A092181 may be interpreted as 5-dimensional icositetrachoronal hyperpyramidal numbers. The icositetrachoron is a convex regular 4-D polytope with Schlaefli symbol {3,4,3}.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A092181.

Table[n*(7 - 10*n^2 + 15*n^3 + 18*n^4)/30, {n, 40}] (* Wesley Ivan Hurt, Oct 30 2022 *)

Vec(x*(1 + 19*x + 43*x^2 + 9*x^3) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Aug 15 2018

a(n) = (n*(7 - 10*n^2 + 15*n^3 + 18*n^4)) / 30 \\ Colin Barker, Aug 15 2018

a(n) = Sum_{k=1..n} A092181(k).
From Colin Barker, Aug 15 2018: (Start)
G.f.: x*(1 + 19*x + 43*x^2 + 9*x^3) / (1 - x)^6.
a(n) = n*(7 - 10*n^2 + 15*n^3 + 18*n^4) / 30.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A302561 Partial sums of A092182.

Original entry on oeis.org

1, 121, 1068, 4720, 14705, 36981, 80416, 157368, 284265, 482185, 777436, 1202136, 1794793, 2600885, 3673440, 5073616, 6871281, 9145593, 11985580, 15490720, 19771521, 24950101, 31160768, 38550600, 47280025, 57523401, 69469596, 83322568, 99301945

Offset: 1

Alejandro J. Becerra Jr., Aug 15 2018

Geometrically, the partial sums of A092182 may be interpreted as 5-dimensional hexacosichoronal hyperpyramidal numbers. The hexacosichoron is a convex regular 4-D polytope with Schlaefli symbol {3,3,5}.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A092182.

Accumulate[LinearRecurrence[{5,-10,10,-5,1},{1,120,947,3652,9985},30]] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{1,121,1068,4720,14705,36981},30] (* Harvey P. Dale, May 04 2024 *)

Vec(x*(1 + 115*x + 357*x^2 + 107*x^3) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Aug 15 2018

a(n) = (n*(12 + n - 64*n^2 + 5*n^3 + 58*n^4)) / 12 \\ Colin Barker, Aug 15 2018

a(n) = Sum_{k=1..n} A092182(k).
From Colin Barker, Aug 15 2018: (Start)
G.f.: x*(1 + 115*x + 357*x^2 + 107*x^3) / (1 - x)^6.
a(n) = (n*(12 + n - 64*n^2 + 5*n^3 + 58*n^4)) / 12.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
(End)

A302560 Partial sums of icosahedral numbers (A006564).

Original entry on oeis.org

1, 13, 61, 185, 440, 896, 1638, 2766, 4395, 6655, 9691, 13663, 18746, 25130, 33020, 42636, 54213, 68001, 84265, 103285, 125356, 150788, 179906, 213050, 250575, 292851, 340263, 393211, 452110, 517390, 589496, 668888, 756041, 851445, 955605, 1069041, 1192288, 1325896

Offset: 1

Alejandro J. Becerra Jr., Aug 15 2018

Geometrically, the partial sums of A006564 may be interpreted as 4-dimensional icosahedral hyperpyramidal numbers.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A006564.

Vec(x*(1 + 8*x + 6*x^2) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Aug 15 2018

a(n) = (n*(2 - 3*n + 10*n^2 + 15*n^3)) / 24 \\ Colin Barker, Aug 15 2018

a(n) = Sum_{k=1..n} A006564(k).
From Colin Barker, Aug 15 2018: (Start)
G.f.: x*(1 + 8*x + 6*x^2) / (1 - x)^5.
a(n) = n*(2 - 3*n + 10*n^2 + 15*n^3)/24.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)

A302559 Partial sums of A092183.

Original entry on oeis.org

1, 601, 5584, 25052, 78557, 198233, 431928, 846336, 1530129, 2597089, 4189240, 6479980, 9677213, 14026481, 19814096, 27370272, 37072257, 49347465, 64676608, 83596828, 106704829, 134660009, 168187592, 208081760, 255208785, 310510161

Offset: 1

Alejandro J. Becerra Jr., Aug 15 2018

Geometrically, the partial sums of A092183 may be interpreted as 5-dimensional hecatonicosachoronal hyperpyramidal numbers. The hecatonicosachoron is a convex regular 4-D polytope with Schlaefli symbol {5,3,3}.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A092183.

Vec(x*(1 + 595*x + 1993*x^2 + 543*x^3) / (1 - x)^6 + O(x^40)) \\ Colin Barker, Aug 15 2018

a(n) = (n*(584 - 105*n - 2120*n^2 + 135*n^3 + 1566*n^4)) / 60 \\ Colin Barker, Aug 15 2018

a(n) = Sum_{k=1..n} A092183(k).
From Colin Barker, Aug 15 2018: (Start)
G.f.: x*(1 + 595*x + 1993*x^2 + 543*x^3) / (1 - x)^6.
a(n) = n*(584 - 105*n - 2120*n^2 + 135*n^3 + 1566*n^4)/60.
a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6. (End)

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User: Alejandro J. Becerra Jr.

A318159 Figurate numbers based on the small stellated dodecahedron: a(n) = n*(21*n^2 - 33*n + 14)/2.

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A300624 Figurate numbers based on the 11-dimensional regular convex polytope called the 11-dimensional cross-polytope, or 11-dimensional hyperoctahedron.

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A302562 Partial sums of A092181.

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A302561 Partial sums of A092182.

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A302560 Partial sums of icosahedral numbers (A006564).

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A302559 Partial sums of A092183.

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A318159 Figurate numbers based on the small stellated dodecahedron: a(n) = n(21n^2 - 33*n + 14)/2.