A193248 Truncated dodecahedron, and truncated icosahedron with faces of centered polygons.

1, 93, 455, 1267, 2709, 4961, 8203, 12615, 18377, 25669, 34671, 45563, 58525, 73737, 91379, 111631, 134673, 160685, 189847, 222339, 258341, 298033, 341595, 389207, 441049, 497301, 558143, 623755, 694317, 770009, 851011, 937503, 1029665, 1127677, 1231719

Offset: 1

Author

Craig Ferguson, Jul 19 2011

Keywords

• nonn
• easy

Comments

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a truncated dodecahedron or truncated icosahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered triangles (A005448) and centered decagons (A062786)] & [centered hexagons (A003215) and centered pentagons (A005891)] respectively.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000
• 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. A003215, A005448, A005891, A062786.

Programs

Magma

[30*n^3-45*n^2+17*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011

Mathematica

Table[30n^3-45n^2+17n-1,{n,40}] (* or *) LinearRecurrence[{4,-6,4,-1},{1,93,455,1267},40] (* Harvey P. Dale, Aug 28 2011 *)

PARI

vector(40, n, 30*n^3 - 45*n^2 + 17*n - 1) \\ G. C. Greubel, Nov 10 2018

Formula

a(n) = 30*n^3 - 45*n^2 + 17*n - 1.

G.f.: x*(1+x)*(x^2 + 88*x + 1) / (x-1)^4. - R. J. Mathar, Aug 26 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(1)=1, a(2)=93, a(3)=455, a(4)=1267. - Harvey P. Dale, Aug 28 2011

E.g.f.: 1 - (1 - 2*x - 45*x^2 - 30*x^3)*exp(x). - G. C. Greubel, Nov 10 2018