%I A057088
%S A057088 1,5,30,175,1025,6000,35125,205625,1203750,7046875,41253125,241500000,
%T A057088 1413765625,8276328125,48450468750,283633984375,1660422265625,
%U A057088 9720281250000,56903517578125,333118994140625,1950112558593750
%N A057088 Scaled Chebyshev U-polynomials evaluated at i*sqrt(5)/2. Generalized 
               Fibonacci sequence.
%C A057088 a(n) gives the length of the word obtained after n steps with the substitution 
               rule 0->11111, 1->111110, starting from 0. The number of 1's and 
               0's of this word is 5*a(n-1) and 5*a(n-2), resp.
%D A057088 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=5, q=5.
%D A057088 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=5.
%H A057088 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057088 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057088 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057088 a(n) = 5*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057088 a(n)= S(n, i*sqrt(5))*(-i*sqrt(5))^n with S(n, x) := U(n, x/2), Chebyshev's 
               polynomials of the 2nd kind, A049310.
%F A057088 G.f.: 1/(1-5*x-5*x^2).
%F A057088 a(n)=(1/3)*sum(k=0, n, binomial(n, k)*Fibonacci(k)*3^k) - Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Oct 25 2003
%F A057088 a(n)=((5+3sqrt(5))/2)^n(1/2+sqrt(5)/6)+(1/2-sqrt(5)/6)((5-3sqrt(5))/2)^n 
               - Paul Barry (pbarry(AT)wit.ie), Sep 22 2004
%F A057088 (a(n)) appears to be given by the floretion - 0.75'i - 0.5'j + 'k - 0.75i' 
               + 0.5j' + 0.5k' + 1.75'ii' - 1.25'jj' + 1.75'kk' - 'ij' - 0.5'ji' 
               - 0.75'jk' - 0.75'kj' - 1.25e ("jes") - Creighton Dement (Smith(AT)xxx.yyy.com), 
               Nov 28 2004
%F A057088 a(n)=Sum_{k, 0<=k<=n}4^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%p A057088 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=5*a[n-1]+5*a[n-2]od: seq(a[n], 
               n=1..33);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 
               14 2008]
%o A057088 (Other) sage: [lucas_number1(n,5,-5) for n in xrange(1, 22)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
%Y A057088 Sequence in context: A094972 A084158 A111469 this_sequence A156195 A105481 
               A094167
%Y A057088 Adjacent sequences: A057085 A057086 A057087 this_sequence A057089 A057090 
               A057091
%K A057088 nonn,easy
%O A057088 0,2
%A A057088 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000