A193250 Small rhombicuboctahedron with faces of centered polygons.

1, 51, 245, 679, 1449, 2651, 4381, 6735, 9809, 13699, 18501, 24311, 31225, 39339, 48749, 59551, 71841, 85715, 101269, 118599, 137801, 158971, 182205, 207599, 235249, 265251, 297701, 332695, 370329, 410699, 453901, 500031, 549185, 601459, 656949, 715751

Offset

1,2

Comments

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a small rhombicuboctahedron. 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 squares (A001844)]

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) = 16*n^3 - 24*n^2 + 10*n - 1.

G.f.: x*(1+x)*(x^2 + 46*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)=51, a(2)=245, a(3)=679. - Harvey P. Dale, Aug 27 2011

E.g.f.: 1 + (-1 + 2*x + 24*x^2 + 16*x^3)*exp(x). - G. C. Greubel, Nov 10 2018

Mathematica

Table[16n^3-24n^2+10n-1, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 51, 245, 679}, 40] (* Harvey P. Dale, Aug 27 2011 *)

Prog

(Excel) copy and paste the following formula   =16 *ROW()^3-24 *ROW()^2+10*ROW()-1 fill down to desired size.
(Magma) [16*n^3-24*n^2+10*n-1: n in [1..50]]; // Vincenzo Librandi, Aug 30 2011
(PARI) vector(40, n, 16*n^3-24*n^2+10*n-1) \\ G. C. Greubel, Nov 10 2018

Crossrefs

Cf. A079414 (partial sums).

Cf. A005448, A001844.

Sequence in context: A107253 A030535 A201813 * A204717 A253921 A204956

Adjacent sequences: A193247 A193248 A193249 * A193251 A193252 A193253

Keyword

nonn,easy

Author

Craig Ferguson, Jul 19 2011

Status

approved