Search: id:A057084
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%I A057084
%S A057084 1,8,56,384,2624,17920,122368,835584,5705728,38961152,266043392,
%T A057084 1816657920,12404916224,84706066432,578409201664,3949625081856,
%U A057084 26969727041536,184160815677440,1257528709087232,8586943147278336
%N A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2).
%D A057084 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=8, q=-8.
%D A057084 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(38) and 
               (45),lhs, m=8.
%H A057084 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057084 a(n) = 8*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.
%F A057084 a(n)= S(n, 2*sqrt(2))*(2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's 
               polynomials of the 2nd kind, A049310.
%F A057084 a(2*k)= A002315(k)*8^k, a(2*k+1)=A001109(k+1)*8^(k+1).
%F A057084 G.f.: 1/(1-8*x+8*x^2).
%F A057084 a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*8^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 28 2008]
%F A057084 a(n)=-(1/8)*[4-2*sqrt(2)]^(n+1)*sqrt(2)+(1/8)*sqrt(2)*[4+2*sqrt(2)]^(n+1), 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
%F A057084 ((4+sqrt8)^n-(4-sqrt8)^n)/sqrt32. Offset 1. a(3)=56. [From Al Hakanson 
               (hawkuu(AT)gmail.com), Jan 07 2009]
%F A057084 a(n)= fourth binomial transform of 1,4,8,32,64,256,512 [From Al Hakanson 
               (hawkuu(AT)gmail.com), Aug 15 2009]
%o A057084 (Other) sage: [lucas_number1(n,8,8) for n in xrange(1, 21)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
%Y A057084 Sequence in context: A034006 A081626 A003494 this_sequence A101596 A092521 
               A156088
%Y A057084 Adjacent sequences: A057081 A057082 A057083 this_sequence A057085 A057086 
               A057087
%K A057084 nonn,easy
%O A057084 0,2
%A A057084 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000

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