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A100145 Structured great rhombicosidodecahedral numbers.

%I A100145 #28 Jun 09 2025 10:26:13
%S A100145 1,120,579,1600,3405,6216,10255,15744,22905,31960,43131,56640,72709,
%T A100145 91560,113415,138496,167025,199224,235315,275520,320061,369160,423039,
%U A100145 481920,546025,615576,690795,771904,859125,952680,1052791,1159680
%N A100145 Structured great rhombicosidodecahedral numbers.
%C A100145 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.
%H A100145 Vincenzo Librandi, <a href="/A100145/b100145.txt">Table of n, a(n) for n = 1..5000</a>
%H A100145 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A100145 a(n) = (1/6)*(222*n^3 - 312*n^2 + 96*n).
%F A100145 From _Jaume Oliver Lafont_, Sep 08 2009: (Start)
%F A100145 a(n) = (1+(n-1))*(1+22*(n-1)+37*(n-1)^2);
%F A100145 G.f.: x*(1+116*x+105*x^2)/(1-x)^4. (End)
%F A100145 E.g.f.: exp(x)*x*(1 + 59*x + 37*x^2). - _Stefano Spezia_, Jun 06 2025
%t A100145 LinearRecurrence[{4,-6,4,-1},{1,120,579,1600},40] (* _Harvey P. Dale_, Mar 27 2019 *)
%o A100145 (Magma) [(1+(n-1))*(1+22*(n-1)+37*(n-1)^2): n in [1..40]]; // _Vincenzo Librandi_, Jul 19 2011
%o A100145 (PARI) a(n)=37*n^3 - 52*n^2 + 16*n \\ _Charles R Greathouse IV_, Nov 07 2016
%Y A100145 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.
%K A100145 easy,nonn
%O A100145 1,2
%A A100145 James A. Record (james.record(AT)gmail.com), Nov 07 2004
%E A100145 Corrected by _T. D. Noe_, Oct 25 2006