%I A057092
%S A057092 1,9,90,891,8829,87480,866781,8588349,85096170,843160671,8354311569,
%T A057092 82777250160,820184055561,8126651751489,80521522263450,797833566134451,
%U A057092 7905195795581109,78327264255440040,776092140459190341
%N A057092 Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci 
               sequence.
%C A057092 a(n) gives the length of the word obtained after n steps with the substitution 
               rule 0->1^9, 1->(1^9)0, starting from 0. The number of 1's and 0's 
               of this word is 9*a(n-1) and 9*a(n-2), resp.
%D A057092 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=9, q=9.
%D A057092 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=9.
%H A057092 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057092 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057092 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057092 a(n) = 9*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057092 a(n)= S(n, i*3)*(-i*3)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials 
               of the 2nd kind, A049310.
%F A057092 G.f.: 1/(1-9*x-9*x^2).
%F A057092 a(n)=Sum_{k, 0<=k<=n}8^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A057092 a(n)=(1/39)*[(9/2)+(3/2)*sqrt(13)]^(n+1)*sqrt(13)-(1/39)*sqrt(13)*[(9/
               2)-(3/2)*sqrt(13)]^(n+1), with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), 
               Nov 20 2008]
%o A057092 (Other) sage: [lucas_number1(n,9,-9) for n in xrange(1, 20)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
%Y A057092 Sequence in context: A054616 A052386 A158609 this_sequence A156577 A052268 
               A155199
%Y A057092 Adjacent sequences: A057089 A057090 A057091 this_sequence A057093 A057094 
               A057095
%K A057092 nonn,easy
%O A057092 0,2
%A A057092 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000