A193253 Great rhombicosidodecahedron with faces of centered polygons.

1, 183, 905, 2527, 5409, 9911, 16393, 25215, 36737, 51319, 69321, 91103, 117025, 147447, 182729, 223231, 269313, 321335, 379657, 444639, 516641, 596023, 683145, 778367, 882049, 994551, 1116233, 1247455, 1388577, 1539959, 1701961, 1874943, 2059265, 2255287

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 edges and n-1 vertices to complete the centered polygon of each face.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000.

OEIS Wiki, (Centered_polygons) pyramidal numbers.

Eric W. Weisstein, MathWorld: Great Rhombicosidodecahedron.

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) = 60*n^3 - 90*n^2 + 32*n - 1.

G.f.: x*(1 + 179*x + 179*x^2 + x^3)/(1-x)^4 = x*(1+x)*(1 + 178*x + x^2)/(1-x)^4. - Colin Barker, Feb 12 2012

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {1, 183, 905, 2527}, 50] (* Vincenzo Librandi, Feb 18 2012 *)
a[n_]:=60*n^3 - 90*n^2 + 32*n - 1 ; Array[a, 50] (* or *)
CoefficientList[Series[(1 + x)*(1 + 178*x + x^2)/(1 - x)^4 , {x, 0, 50}], x] (* Stefano Spezia, Sep 02 2018 *)

Prog

(Excel) =60*ROW()^3-90*ROW()^2+32*ROW()-1 fill down  to desired size.
(PARI) a(n)=60*n^3-90*n^2+32*n-1 \\ Charles R Greathouse IV, Feb 12 2012
(Magma) [60*n^3-90*n^2+32*n-1: n in [1..40]]; // Vincenzo Librandi, Feb 18 2012

Crossrefs

Cf. A001844 (centered squares), A062786 (centered decagons), and A003215 (centered hexagons).

Sequence in context: A261110 A260172 A252070 * A272126 A171563 A347504

Adjacent sequences: A193250 A193251 A193252 * A193254 A193255 A193256

Keyword

nonn,easy

Author

Craig Ferguson, Jul 19 2011

Status

approved