A100145 Structured great rhombicosidodecahedral numbers.

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

Offset: 1

Author

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

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A051673, A100146 through A100156 - structured Archimedean solids; A100157 through A100175 - structured Catalan solids; A100147 - structured prisms; A000447 - structured diamonds; A100185 - structured anti-prisms; and A100188 - structured anti-diamonds.

Programs

Magma

[(1+(n-1))*(1+22*(n-1)+37*(n-1)^2): n in [1..40]]; // Vincenzo Librandi, Jul 19 2011

Mathematica

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

PARI

a(n)=37*n^3 - 52*n^2 + 16*n \\ Charles R Greathouse IV, Nov 07 2016

Formula

a(n) = (1/6)*(222*n^3 - 312*n^2 + 96*n).

From Jaume Oliver Lafont, Sep 08 2009: (Start)

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)

E.g.f.: exp(x)*x*(1 + 59*x + 37*x^2). - Stefano Spezia, Jun 06 2025

Extensions

Corrected by T. D. Noe, Oct 25 2006