A193228 Truncated octahedron with faces of centered polygons.

1, 39, 185, 511, 1089, 1991, 3289, 5055, 7361, 10279, 13881, 18239, 23425, 29511, 36569, 44671, 53889, 64295, 75961, 88959, 103361, 119239, 136665, 155711, 176449, 198951, 223289, 249535, 277761, 308039, 340441, 375039, 411905, 451111, 492729, 536831, 583489

Offset

1,2

Comments

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a truncated octahedron. Each iteration requires the addition of (n-2) edges and (n-1) vertices to complete the centered polygon of each face. [centered squares (A001844) and centered hexagons (A003215)]

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) = 12*n^3 - 18*n^2 + 8*n - 1.

G.f.: x*(1+x)*(x^2 + 34*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(0)=1, a(1)=39, a(2)=185, a(3)=511. - Harvey P. Dale, Aug 27 2011

E.g.f.: 1 - (1 - 2*x - 18*x^2 - 12*x^3)*exp(x). - G. C. Greubel, Nov 10 2018

Mathematica

Table[12n^3-18n^2+8n-1, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 39, 185, 511}, 40] (* Harvey P. Dale, Aug 27 2011 *)

Prog

(Excel) (copy and paste the following formula =12*ROW()^3-18*ROW()^2+8*ROW()-1 fill down to desired size.)
(Magma) [12*n^3-18*n^2+8*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011
(PARI) vector(40, n, 12*n^3 - 18*n^2 + 8*n - 1) \\ G. C. Greubel, Nov 10 2018

Crossrefs

Sequence in context: A235981 A235974 A258095 * A243578 A124619 A290071

Adjacent sequences: A193225 A193226 A193227 * A193229 A193230 A193231

Keyword

nonn,easy

Author

Craig Ferguson, Jul 18 2011

Status

approved