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A100145
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Structured great rhombicosidodecahedral numbers.
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64
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1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, 56640, 72709, 91560, 113415, 138496, 167025, 199224, 235315, 275520, 320061, 369160, 423039, 481920, 546025, 615576, 690795, 771904, 859125, 952680, 1052791, 1159680
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OFFSET
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1,2
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COMMENTS
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Structured polyhedral numbers are a type of figurate polyhedral numbers. Structurate polyhedra differ from regular figurate polyhedra by having appropriate figurate polygonal faces at any iteration, i.e., a regular truncated octahedron, n=2, would have 7 points on its hexagonal faces, whereas a structured truncated octahedron, n=2, would have 6 points - just as a hexagon, n=2, would have. Like regular figurate polygons, structured polyhedra seem to originate at a vertex and since many polyhedra have different vertices (a pentagonal diamond has 2 "polar" vertices with 5 adjacent vertices and 5 "equatorial" vertices with 4 adjacent vertices), these polyhedra have multiple structured number sequences, dependent on the "vertex structures" which are each equal to the one vertex itself plus its adjacent vertices. For polystructurate polyhedra the notation, structured polyhedra (vertex structure x) is used to differentiate between alternate vertices, where VS stands for vertex structure.
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LINKS
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FORMULA
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a(n) = (1/6)*(222*n^3 - 312*n^2 + 96*n).
a(n) = (1+(n-1))*(1+22*(n-1)+37*(n-1)^2);
g.f.: x*(1+116*x+105*x^2)/(1-x)^4. (End)
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MATHEMATICA
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LinearRecurrence[{4, -6, 4, -1}, {1, 120, 579, 1600}, 40] (* Harvey P. Dale, Mar 27 2019 *)
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PROG
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(Magma) [(1+(n-1))*(1+22*(n-1)+37*(n-1)^2): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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James A. Record (james.record(AT)gmail.com), Nov 07 2004
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EXTENSIONS
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STATUS
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approved
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