

A100145


Structured great rhombicosidodecahedral numbers.


64



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

1,2


COMMENTS

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.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,6,4,1).


FORMULA

a(n) = (1/6)*(222*n^3  312*n^2 + 96*n).
From Jaume Oliver Lafont, Sep 08 2009: (Start)
a(n) = (1+(n1))*(1+22*(n1)+37*(n1)^2);
g.f.: x*(1+116*x+105*x^2)/(1x)^4. (End)


MATHEMATICA

LinearRecurrence[{4, 6, 4, 1}, {1, 120, 579, 1600}, 40] (* Harvey P. Dale, Mar 27 2019 *)


PROG

(MAGMA) [(1+(n1))*(1+22*(n1)+37*(n1)^2): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011
(PARI) a(n)=37*n^3  52*n^2 + 16*n \\ Charles R Greathouse IV, Nov 07 2016


CROSSREFS

Cf. A051673, A100146 through A100156  structured Archimedean solids; A100157 through A100175  structured Catalan solids; A100147  structured prisms; A000447  structured diamonds; A100185  structured antiprisms; and A100188  structured antidiamonds.
Sequence in context: A241613 A115619 A152622 * A234437 A090216 A265041
Adjacent sequences: A100142 A100143 A100144 * A100146 A100147 A100148


KEYWORD

easy,nonn


AUTHOR

James A. Record (james.record(AT)gmail.com), Nov 07 2004


EXTENSIONS

Corrected by T. D. Noe, Oct 25 2006


STATUS

approved



