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Search: id:A057090
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| A057090 |
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Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence. |
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+0
5
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| 1, 7, 56, 441, 3479, 27440, 216433, 1707111, 13464808, 106203433, 837677687, 6607167840, 52113918689, 411047605703, 3242130670744, 25572247935129, 201700650241111, 1590910287233680, 12548276562323537, 98974307946900519
(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->1^7, 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.
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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=7, q=7.
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.
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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) = 7*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
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.
G.f.: 1/(1-7*x-7*x^2).
a(n)=Sum_{k, 0<=k<=n}6^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2006
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]
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PROGRAM
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(Other) sage: [lucas_number1(n, 7, -7) for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 24 2009]
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CROSSREFS
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Sequence in context: A122996 A092315 A092318 this_sequence A156362 A055274 A152776
Adjacent sequences: A057087 A057088 A057089 this_sequence A057091 A057092 A057093
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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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