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A193253 Great rhombicosidodecahedron with faces of centered polygons. 1

%I

%S 1,183,905,2527,5409,9911,16393,25215,36737,51319,69321,91103,117025,

%T 147447,182729,223231,269313,321335,379657,444639,516641,596023,

%U 683145,778367,882049,994551,1116233,1247455,1388577,1539959,1701961,1874943,2059265,2255287

%N Great rhombicosidodecahedron with faces of centered polygons.

%C 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.

%H Vincenzo Librandi, <a href="/A193253/b193253.txt">Table of n, a(n) for n = 1..1000</a>.

%H OEIS Wiki, <a href="http://oeis.org/wiki/(Centered_polygons)_pyramidal_numbers"> (Centered_polygons) pyramidal numbers</a>.

%H Eric W. Weisstein, <a href="http://mathworld.wolfram.com/GreatRhombicosidodecahedron.html">MathWorld: Great Rhombicosidodecahedron</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetrahedral_number">Tetrahedral number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Triangular_number">Triangular number</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Centered_polygonal_number">Centered polygonal number</a>.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F a(n) = 60*n^3 - 90*n^2 + 32*n - 1.

%F 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

%t LinearRecurrence[{4, -6, 4, -1}, {1, 183, 905, 2527}, 50] (* _Vincenzo Librandi_, Feb 18 2012 *)

%t a[n_]:=60*n^3 - 90*n^2 + 32*n - 1 ; Array[a, 50] (* or *)

%t CoefficientList[Series[(1 + x)*(1 + 178*x + x^2)/(1 - x)^4 , {x, 0, 50}], x] (* _Stefano Spezia_, Sep 02 2018 *)

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

%o (PARI) a(n)=60*n^3-90*n^2+32*n-1 \\ _Charles R Greathouse IV_, Feb 12 2012

%o (MAGMA) [60*n^3-90*n^2+32*n-1: n in [1..40]] // _Vincenzo Librandi_, Feb 18 2012

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

%K nonn,easy

%O 1,2

%A _Craig Ferguson_, Jul 19 2011

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Last modified July 21 19:13 EDT 2019. Contains 325199 sequences. (Running on oeis4.)