A193251 Small rhombicosidodecahedron with faces of centered polygons.

1, 123, 605, 1687, 3609, 6611, 10933, 16815, 24497, 34219, 46221, 60743, 78025, 98307, 121829, 148831, 179553, 214235, 253117, 296439, 344441, 397363, 455445, 518927, 588049, 663051, 744173, 831655, 925737, 1026659, 1134661, 1249983, 1372865, 1503547, 1642269

Offset: 1

Author

Craig Ferguson, Jul 19 2011

Keywords

• nonn
• easy
• changed

Comments

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a small rhombicosidodecahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448), centered squares (A001844) and centered pentagons (A005891)]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• OEIS Wiki, (Centered_polygons) pyramidal numbers.
• Wikipedia, Tetrahedral number.
• Wikipedia, Triangular number.
• Wikipedia, Centered polygonal number.
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A001844, A005448, A005891, A060747, A195317.

Programs

Magma

[40*n^3-60*n^2+22*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

Mathematica

A193251[n_] := (2*n - 1)*(20*(n - 1)*n + 1); Array[A193251, 50] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 123, 605, 1687}, 50] (* Paolo Xausa, Aug 25 2025 *)

PARI

a(n)=40*n^3-60*n^2+22*n-1 \\ Charles R Greathouse IV, Oct 19 2022

Formula

a(n) = 40*n^3 - 60*n^2 + 22*n - 1.

G.f.: x*(1+x)*(x^2 + 118*x + 1)/(x-1)^4. - R. J. Mathar, Aug 26 2011

From Elmo R. Oliveira, Aug 22 2025: (Start)

E.g.f.: 1 + exp(x)*(-1 + 2*x + 60*x^2 + 40*x^3).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 4.

a(n) = A060747(n)*A195317(n). (End)