A193252 Great rhombicuboctahedron with faces of centered polygons.

1, 75, 365, 1015, 2169, 3971, 6565, 10095, 14705, 20539, 27741, 36455, 46825, 58995, 73109, 89311, 107745, 128555, 151885, 177879, 206681, 238435, 273285, 311375, 352849, 397851, 446525, 499015, 555465, 616019, 680821, 750015, 823745, 902155, 985389, 1073591

Offset

1,2

Comments

The sequence starts with a central dot and expands outward with (n-1) centered polygonal pyramids producing a great rhombicosidodecahedron. Each iteration requires the addition of (n-2) edge units and (n-1) vertices to complete the centered polygon of each face: centered squares, centered octagons and centered hexagons.

Links

Bruno Berselli, Table of n, a(n) for n = 1..1000

OEIS Wiki, (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) = 24*n^3 - 36*n^2 + 14*n - 1.

G.f.: x*(1+x)*(1+70*x+x^2)/(1-x)^4; a(n) = (2*n-1)*A069190(n). - Bruno Berselli, Jul 21 2011

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4); a(0)=1, a(1)=75, a(2)=365, a(3)=1015. - Harvey P. Dale, Jul 27 2011

a(n) = 72 * A000330(n-1) + A005408(n-1). - Bruce J. Nicholson, Feb 23 2019

E.g.f.: 1 + (-1 + 2*x + 36*x^2 + 24*x^3)*exp(x). - G. C. Greubel, Feb 26 2019

Mathematica

Table[24n^3-36n^2+14n-1, {n, 40}] (* or *) LinearRecurrence[{4, -6, 4, -1}, {1, 75, 365, 1015}, 40] (* Harvey P. Dale, Jul 27 2011 *)

Prog

(Excel) =24*ROW()^3-36*ROW()^2+14*ROW()-1
(Magma) A069190:=func<n | 12*n^2-12*n+1>; [(2*n-1)*A069190(n): n in [1..40]];  // Bruno Berselli, Jul 21 2011
(PARI) for(n=1, 40, print1(24*n^3-36*n^2+14*n-1", "));  \\ Bruno Berselli, Jul 21 2011
(Sage) [24*n^3 -36*n^2 +14*n -1 for n in (1..40)] # G. C. Greubel, Feb 26 2019
(GAP) List([1..40], n-> 24*n^3 -36*n^2 +14*n -1) # G. C. Greubel, Feb 26 2019

Crossrefs

First differences in 2*A158591.

Cf. A001844 (centered square numbers), A016754 (centered octagonal numbers), A003215 (centered hexagonal numbers).

Sequence in context: A223078 A055561 A350245 * A223452 A015223 A129625

Adjacent sequences: A193249 A193250 A193251 * A193253 A193254 A193255

Keyword

nonn,easy

Author

Craig Ferguson, Jul 19 2011

Status

approved