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Search: id:A057091
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| A057091 |
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence. |
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+0
6
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| 1, 8, 72, 640, 5696, 50688, 451072, 4014080, 35721216, 317882368, 2828828672, 25173688320, 224020135936, 1993550594048, 17740565839872, 157872931471360, 1404907978489856, 12502247279689728, 111257242065436672
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^8, 1->(1^8)0, starting from 0. The number of 1's and 0's of this word is 8*a(n-1) and 8*a(n-2), resp.
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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.(39) and (45),rhs, m=8.
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LINKS
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Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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, i*2*sqrt(2))*(-i*2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
G.f.: 1/(1-8*x-8*x^2).
a(n)=Sum_{k, 0<=k<=n}7^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
a(n)=-(1/6)*sqrt(6)*[4-2*sqrt(6)]^n+(1/2)*[4+2*sqrt(6)]^n+(1/6)*[4+2*sqrt(6)]^n*sqrt(6)+(1/2) *[4-2*sqrt(6)]^n, with n>=0 - Paolo P. Lava (ppl(AT)spl.at), Jul 08 2008
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PROGRAM
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(Other) sage: [lucas_number1(n, 8, -8) for n in xrange(0, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 25 2009]
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CROSSREFS
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Sequence in context: A052379 A158798 A062541 this_sequence A156566 A055275 A155198
Adjacent sequences: A057088 A057089 A057090 this_sequence A057092 A057093 A057094
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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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