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A318159
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Figurate numbers based on the small stellated dodecahedron: a(n) = n*(21*n^2 - 33*n + 14)/2.
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1
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1, 32, 156, 436, 935, 1716, 2842, 4376, 6381, 8920, 12056, 15852, 20371, 25676, 31830, 38896, 46937, 56016, 66196, 77540, 90111, 103972, 119186, 135816, 153925, 173576, 194832, 217756, 242411, 268860, 297166, 327392, 359601, 393856, 430220, 468756, 509527
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OFFSET
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1,2
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COMMENTS
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The small stellated dodecahedron is a 3D nonconvex regular polyhedron represented by the Schlaefli symbol {5/2, 5}.
When truncated, a degenerate dodecahedron is produced. It is then easy to recognize that every small stellated dodecahedron can be constructed by morphing the 12 pentagonal faces of a regular dodecahedron into pentagonal pyramids.
The last digits form a cycle of length 20 [1, 2, 6, 6, ..., 1, 2, 6, 6].
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LINKS
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FORMULA
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a(n) == a(n+20) (mod 10).
G.f.: x*(1 + 28*x + 34*x^2) / (1 - x)^4.
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
(End)
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MATHEMATICA
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CoefficientList[Series[(1 + 28*x + 34*x^2) / (1 - x)^4 , {x, 0, 45}], x] (* or *)
LinearRecurrence[{4, -6, 4, -1}, {1, 32, 156, 436}, 45] (* Stefano Spezia, Sep 02 2018 *)
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PROG
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(PARI) Vec(x*(1 + 28*x + 34*x^2) / (1 - x)^4 + O(x^40)) \\ Colin Barker, Aug 20 2018
(PARI) a(n) = (n*(14 - 33*n + 21*n^2)) / 2 \\ Colin Barker, Aug 20 2018
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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