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Search: id:A057084
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| A057084 |
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Scaled Chebyshev U-polynomials evaluated at sqrt(2). |
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+0
8
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| 1, 8, 56, 384, 2624, 17920, 122368, 835584, 5705728, 38961152, 266043392, 1816657920, 12404916224, 84706066432, 578409201664, 3949625081856, 26969727041536, 184160815677440, 1257528709087232, 8586943147278336
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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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.
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.
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LINKS
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Index entries for sequences related to Chebyshev polynomials.
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FORMULA
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a(n) = 8*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.
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.
a(2*k)= A002315(k)*8^k, a(2*k+1)=A001109(k+1)*8^(k+1).
G.f.: 1/(1-8*x+8*x^2).
a(n)=Sum_{k, 0<=k<=n} A109466(n,k)*8^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 28 2008]
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]
((4+sqrt8)^n-(4-sqrt8)^n)/sqrt32. Offset 1. a(3)=56. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 07 2009]
a(n)= fourth binomial transform of 1,4,8,32,64,256,512 [From Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009]
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PROGRAM
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(Other) sage: [lucas_number1(n, 8, 8) for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 23 2009]
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CROSSREFS
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Sequence in context: A034006 A081626 A003494 this_sequence A101596 A092521 A156088
Adjacent sequences: A057081 A057082 A057083 this_sequence A057085 A057086 A057087
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000
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