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A193250 Small rhombicuboctahedron with faces of centered polygons. 1
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 (list; graph; refs; listen; history; text; internal format)
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

Sequence in context: A107253 A030535 A201813 * A204717 A253921 A204956

Adjacent sequences:  A193247 A193248 A193249 * A193251 A193252 A193253

KEYWORD

nonn

AUTHOR

Craig Ferguson, Jul 19 2011

STATUS

approved

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Last modified July 24 02:56 EDT 2019. Contains 325290 sequences. (Running on oeis4.)