A076765 Partial sums of Chebyshev sequence S(n,8) = U(n,4) = A001090(n+1).

1, 9, 72, 568, 4473, 35217, 277264, 2182896, 17185905, 135304345, 1065248856, 8386686504, 66028243177, 519839258913, 4092685828128, 32221647366112, 253680493100769, 1997222297440041, 15724097886419560, 123795560793916440

Offset: 0

Author

Maurice MARGENSTERN (margens(AT)lita.univ-metz.fr), Nov 14 2002

Keywords

• nice
• easy
• nonn

Comments

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).

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).

References

• 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.
• 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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• M. Margenstern, Number of polyhedra at distance n in {5,3,4}
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (9,-9,1).

Crossrefs

Cf. A092521 (partial sums of S(n, 7)).

Cf. A212336 for more sequences with g.f. of the type 1/(1-k*x+k*x^2-x^3).

Programs

Mathematica

Join[{a=1,b=9},Table[c=8*b-a+1; a=b; b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{9,-9,1},{1,9,72},30] (* Harvey P. Dale, Mar 13 2014 *)
CoefficientList[Series[1/((1 - x) (1 - 8 x + x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

Formula

a(n+3) = 9*a(n+2) - 9*a(n+1) + a(n); initial values: a(0) = 1, a(1) = 9, a(2) = 72

a(n) = Sum_{k=0..n} S(k, 8) with S(k, x) = U(k, x/2) Chebyshev's polynomials of the second kind.

G.f.: 1/((1-x)*(1 - 8*x + x^2)) = 1/(1 - 9*x + 9*x^2 - x^3).

a(n) = 8*a(n-1) - a(n-2) + 1; a(-1)=0, a(0)=1.

a(n) = (S(n+1, 8) - S(n, 8) - 1)/6, n >= 0.

Extensions

Extension and Chebyshev comments from Wolfdieter Lang, Aug 31 2004