%I A057089
%S A057089 1,6,42,288,1980,13608,93528,642816,4418064,30365280,208700064,
%T A057089 1434392064,9858552768,67757668992,465697330560,3200729997312,
%U A057089 21998563967232,151195763787264,1039165966526976,7142170381885440
%N A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized 
               Fibonacci sequence.
%C A057089 a(n) gives the length of the word obtained after n steps with the substitution 
               rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's 
               of this word is 6*a(n-1) and 6*a(n-2), resp.
%D A057089 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=6, q=6.
%D A057089 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=6.
%H A057089 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057089 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057089 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057089 a(n) = 6*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057089 a(n)= S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's 
               polynomials of the 2nd kind, A049310.
%F A057089 G.f.: 1/(1-6*x-6*x^2).
%F A057089 a(n)=Sum_{k, 0<=k<=n}5^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A057089 a(n)=-(1/30)*sqrt(15)*[3-sqrt(15)]^(n+1)+(1/30)*sqrt(15)*[3+sqrt(15)]^(n+1), 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
%o A057089 (Other) sage: [lucas_number1(n,6,-6) for n in xrange(1, 21)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
%Y A057089 Sequence in context: A062310 A105482 A157335 this_sequence A110711 A156361 
               A055272
%Y A057089 Adjacent sequences: A057086 A057087 A057088 this_sequence A057090 A057091 
               A057092
%K A057089 nonn,easy
%O A057089 0,2
%A A057089 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000