0,2

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).

M. Margenstern and G. Skordev, Tools for devising cellular automata in the 3-dimensional hyperbolic space, I - the geometrical part, proceddings 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, proceddings of SCI'2002, Orlando, Florida, Jul 14-18, (2002), vol. XI, 548-552

M. Margenstern, numberof polyhedra at distance n in {5,3,4}

Index entries for sequences relate d to Chebyshev polynomials.

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(S(k, 8), k=0..n) 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.

a(n)=-1/6+(7/12)*[4-sqrt(15)]^n-(3/20)*[4-sqrt(15)]^n*sqrt(15)+(7/12)*[4+sqrt(15)]^n+(3/20) *sqrt(15)*[4+sqrt(15)]^n, with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Jun 25 2008

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

Sequence in context: A045993 A084327 A057085 this_sequence A006634 A129328 A003951

Adjacent sequences: A076762 A076763 A076764 this_sequence A076766 A076767 A076768

nice,easy,nonn

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

Extension and Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 31 2004