

A076765


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


15



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

0,2


COMMENTS

In the tiling {5,3,4} of 3dimensional 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 contextfree (M. Margenstern's theorem, see second reference, above).


REFERENCES

M. Margenstern and G. Skordev, Tools for devising cellular automata in the 3dimensional hyperbolic space, I  the geometrical part, proceedings of SCI'2002, Orlando, Florida, Jul 1418, (2002), vol. XI, 542547 Vol. 100 (1993), pp. 125.
M. Margenstern and G. Skordev, Tools for devising cellular automata in the 3dimensional hyperbolic space, II  the numeric algorithms, proceedings of SCI'2002, Orlando, Florida, Jul 1418, (2002), vol. XI, 548552


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).


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/((1x)*(18*x+x^2)) = 1/(19*x+9*x^2x^3).
a(n) = 8*a(n1)  a(n2) +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)*(4sqrt(15))^n(3/20)*(4sqrt(15))^n*sqrt(15)+(7/12)*(4+sqrt(15))^n+(3/20) *sqrt(15)*(4+sqrt(15))^n.  Paolo P. Lava, Jun 25 2008


MATHEMATICA

Join[{a=1, b=9}, Table[c=8*ba+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 *)


CROSSREFS

Cf. A092521 (partial sums of S(n, 7)).
Cf. A212336 for more sequences with g.f. of the type 1/(1k*x+k*x^2x^3).
Sequence in context: A045993 A084327 A057085 * A006634 A129328 A162960
Adjacent sequences: A076762 A076763 A076764 * A076766 A076767 A076768


KEYWORD

nice,easy,nonn


AUTHOR

Maurice MARGENSTERN (margens(AT)lita.univmetz.fr), Nov 14 2002


EXTENSIONS

Extension and Chebyshev comments from Wolfdieter Lang, Aug 31 2004


STATUS

approved



