Search: id:A057090
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%I A057090
%S A057090 1,7,56,441,3479,27440,216433,1707111,13464808,106203433,837677687,
%T A057090 6607167840,52113918689,411047605703,3242130670744,25572247935129,
%U A057090 201700650241111,1590910287233680,12548276562323537,98974307946900519
%N A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized 
               Fibonacci sequence.
%C A057090 a(n) gives the length of the word obtained after n steps with the substitution 
               rule 0->1^7, 1->(1^7)0, starting from 0. The number of 1's and 0's 
               of this word is 7*a(n-1) and 7*a(n-2), resp.
%D A057090 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=7, q=7.
%D A057090 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=7.
%H A057090 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057090 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057090 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057090 a(n) = 7*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057090 a(n)= S(n, i*sqrt(7))*(-i*sqrt(7))^n with S(n, x) := U(n, x/2), Chebyshev's 
               polynomials of the 2nd kind, A049310.
%F A057090 G.f.: 1/(1-7*x-7*x^2).
%F A057090 a(n)=Sum_{k, 0<=k<=n}6^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A057090 a(n)=-(1/77)*[(7/2)-(1/2)*sqrt(77)]^(n+1)*sqrt(77)+(1/77)*[(7/2)+(1/2)*sqrt(77)]^(n+1)*sqrt(77), 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
%o A057090 (Other) sage: [lucas_number1(n,7,-7) for n in xrange(1, 21)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
%Y A057090 Sequence in context: A122996 A092315 A092318 this_sequence A156362 A055274 
               A152776
%Y A057090 Adjacent sequences: A057087 A057088 A057089 this_sequence A057091 A057092 
               A057093
%K A057090 nonn,easy
%O A057090 0,2
%A A057090 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000

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