A318159 Figurate numbers based on the small stellated dodecahedron: a(n) = n*(21*n^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

Author

Alejandro J. Becerra Jr., Aug 19 2018

Keywords

• nonn
• easy
• changed

Comments

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].

Links

• 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).

Crossrefs

Cf. A002411, A006566.

Programs

Magma

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

Mathematica

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 *)

PARI

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

PARI

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

Formula

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

Extensions

More terms from Colin Barker, Aug 20 2018