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

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 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).

Crossrefs

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

Programs

Excel

=60*ROW()^3-90*ROW()^2+32*ROW()-1 fill down  to desired size.

Magma

[60*n^3-90*n^2+32*n-1: n in [1..40]] // Vincenzo Librandi, Feb 18 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 *)

PARI

a(n)=60*n^3-90*n^2+32*n-1 \\ Charles R Greathouse IV, Feb 12 2012

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