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

1,2

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

OEIS, (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).

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

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 *)

Prog

(Excel) copy and paste the formula   =30 *ROW()^3-45 *ROW()^2+17*ROW()-1 fill down to desired size.
(Magma) [30*n^3-45*n^2+17*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011
(PARI) vector(40, n, 30*n^3 - 45*n^2 + 17*n - 1) \\ G. C. Greubel, Nov 10 2018

Crossrefs

Sequence in context: A237604 A045235 A116240 * A146090 A160174 A238693

Adjacent sequences: A193245 A193246 A193247 * A193249 A193250 A193251

Keyword

nonn,easy

Author

Craig Ferguson, Jul 19 2011

Status

approved