This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A076765 #44 Dec 31 2023 10:12:41
%S A076765 1,9,72,568,4473,35217,277264,2182896,17185905,135304345,1065248856,
%T A076765 8386686504,66028243177,519839258913,4092685828128,32221647366112,
%U A076765 253680493100769,1997222297440041,15724097886419560,123795560793916440
%N A076765 Partial sums of Chebyshev sequence S(n,8) = U(n,4) = A001090(n+1).
%C A076765 In the tiling {5,3,4} of 3-dimensional hyperbolic space, the number of regular dodecahedra with right angles of the n generation which are contained in an eighth of space (intersection of three pairwise perpendicular hyperplanes which are supported by the faces of a dodecahedron at a vertex).
%C A076765 Let beta be the greatest real root of the polynomial which is defined by the above recurrent equation. Consider the representation of positive numbers in the basis beta. Then the language which consists of the maximal representations of positive numbers is neither regular nor context-free (M. Margenstern's theorem, see second reference, above).
%D A076765 M. Margenstern and G. Skordev, Tools for devising cellular automata in the 3-dimensional hyperbolic space, I - the geometrical part, proceedings of SCI'2002, Orlando, Florida, Jul 14-18, (2002), vol. XI, 542-547 Vol. 100 (1993), pp. 1-25.
%D A076765 M. Margenstern and G. Skordev, Tools for devising cellular automata in the 3-dimensional hyperbolic space, II - the numeric algorithms, proceedings of SCI'2002, Orlando, Florida, Jul 14-18, (2002), vol. XI, 548-552
%H A076765 Vincenzo Librandi, <a href="/A076765/b076765.txt">Table of n, a(n) for n = 0..1000</a>
%H A076765 M. Margenstern, <a href="http://lita.sciences.univ-metz.fr/~margens/">Number of polyhedra at distance n in {5,3,4} </a>
%H A076765 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A076765 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-9,1).
%F A076765 a(n+3) = 9*a(n+2) - 9*a(n+1) + a(n); initial values: a(0) = 1, a(1) = 9, a(2) = 72
%F A076765 a(n) = Sum_{k=0..n} S(k, 8) with S(k, x) = U(k, x/2) Chebyshev's polynomials of the second kind.
%F A076765 G.f.: 1/((1-x)*(1 - 8*x + x^2)) = 1/(1 - 9*x + 9*x^2 - x^3).
%F A076765 a(n) = 8*a(n-1) - a(n-2) + 1; a(-1)=0, a(0)=1.
%F A076765 a(n) = (S(n+1, 8) - S(n, 8) - 1)/6, n >= 0.
%t A076765 Join[{a=1,b=9},Table[c=8*b-a+1; a=b; b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 19 2011 *)
%t A076765 LinearRecurrence[{9,-9,1},{1,9,72},30] (* _Harvey P. Dale_, Mar 13 2014 *)
%t A076765 CoefficientList[Series[1/((1 - x) (1 - 8 x + x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 15 2014 *)
%Y A076765 Cf. A092521 (partial sums of S(n, 7)).
%Y A076765 Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).
%K A076765 nice,easy,nonn
%O A076765 0,2
%A A076765 Maurice MARGENSTERN (margens(AT)lita.univ-metz.fr), Nov 14 2002
%E A076765 Extension and Chebyshev comments from _Wolfdieter Lang_, Aug 31 2004