%I A057091
%S A057091 1,8,72,640,5696,50688,451072,4014080,35721216,317882368,2828828672,
%T A057091 25173688320,224020135936,1993550594048,17740565839872,157872931471360,
%U A057091 1404907978489856,12502247279689728,111257242065436672
%N A057091 Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci 
               sequence.
%C A057091 a(n) gives the length of the word obtained after n steps with the substitution 
               rule 0->1^8, 1->(1^8)0, starting from 0. The number of 1's and 0's 
               of this word is 8*a(n-1) and 8*a(n-2), resp.
%D A057091 A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. 
               Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=8.
%D A057091 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(39) and 
               (45),rhs, m=8.
%H A057091 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057091 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057091 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057091 a(n) = 8*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057091 a(n)= S(n, i*2*sqrt(2))*(-i*2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's 
               polynomials of the 2nd kind, A049310.
%F A057091 G.f.: 1/(1-8*x-8*x^2).
%F A057091 a(n)=Sum_{k, 0<=k<=n}7^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A057091 a(n)=-(1/6)*sqrt(6)*[4-2*sqrt(6)]^n+(1/2)*[4+2*sqrt(6)]^n+(1/6)*[4+2*sqrt(6)]^n*sqrt(6)+(1/
               2) *[4-2*sqrt(6)]^n, with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Jul 
               08 2008
%o A057091 (Other) sage: [lucas_number1(n,8,-8) for n in xrange(0, 20)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2009]
%Y A057091 Sequence in context: A052379 A158798 A062541 this_sequence A156566 A055275 
               A155198
%Y A057091 Adjacent sequences: A057088 A057089 A057090 this_sequence A057092 A057093 
               A057094
%K A057091 nonn,easy
%O A057091 0,2
%A A057091 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000