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

Author

Craig Ferguson, Jul 19 2011

Keywords

• nonn
• easy

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).

Crossrefs

First differences in 2*A158591.

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

Programs

Excel

=24*ROW()^3-36*ROW()^2+14*ROW()-1

GAP

List([1..40], n-> 24*n^3 -36*n^2 +14*n -1); # G. C. Greubel, Feb 26 2019

Magma

A069190:=func; [(2*n-1)*A069190(n): n in [1..40]];  // Bruno Berselli, Jul 21 2011

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 *)

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

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