%I A057087
%S A057087 1,4,20,96,464,2240,10816,52224,252160,1217536,5878784,28385280,
%T A057087 137056256,661766144,3195289600,15428222976,74494050304,359689093120,
%U A057087 1736732573696,8385686667264,40489676963840,195501454524416
%N A057087 Scaled Chebyshev U-polynomials evaluated at i. Generalized Fibonacci 
               sequence.
%C A057087 a(n) gives the length of the word obtained after n steps with the substitution 
               rule 0->1111, 1->11110, starting from 0. The number of 1's and 0's 
               of this word is 4*a(n-1) and 4*a(n-2), resp.
%C A057087 Inverse binomial transform of odd Pell bisection A001653. With a leading 
               zero, inverse binomial transform of even Pell bisection A001542, 
               divided by 2. - Paul Barry (pbarry(AT)wit.ie), May 16 2003
%D A057087 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=4, q=4.
%D A057087 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=4.
%H A057087 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057087 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057087 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057087 a(n) = 4*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057087 a(n)= S(n, 2*i)*(-2*i)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials 
               of the 2nd kind, A049310.
%F A057087 G.f.: 1/(1-4*x-4*x^2).
%F A057087 a(n)=Sum_{k, 0<=k<=n}3^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A057087 a(n)=-(1/8)*sqrt(2)*[2-2*sqrt(2)]^(n-1)+(1/8)*[2+2*sqrt(2)]^(n-1)*sqrt(2), 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
%F A057087 ((2+sqrt8)^n-(2-sqrt8)^n)/sqrt32. Offset 1. a(3)=20. [From Al Hakanson 
               (hawkuu(AT)gmail.com), Jan 07 2009]
%F A057087 a(n)=A000129(n+1)*A000079(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Jul 08 2009]
%o A057087 (PARI) a(n)=if(n<0, 0, (2*I)^n*subst(I*poltchebi(n+1)+poltchebi(n),'x,
               -I)/2) /* Michael Somos Sep 16 2005 */
%o A057087 (Other) sage: [lucas_number1(n,4,-4) for n in xrange(1, 23)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%Y A057087 Pairwise sums are in A086347.
%Y A057087 Sequence in context: A094971 A099025 A008353 this_sequence A151254 A098225 
               A073532
%Y A057087 Adjacent sequences: A057084 A057085 A057086 this_sequence A057088 A057089 
               A057090
%K A057087 nonn,easy
%O A057087 0,2
%A A057087 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000