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A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence. 11
1, 7, 56, 441, 3479, 27440, 216433, 1707111, 13464808, 106203433, 837677687, 6607167840, 52113918689, 411047605703, 3242130670744, 25572247935129, 201700650241111, 1590910287233680, 12548276562323537, 98974307946900519, 780658091564568392 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
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.
LINKS
Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
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.
Tanya Khovanova, Recursive Sequences
Wolfdieter Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=7.
FORMULA
a(n) = 7*(a(n-1) + a(n-2)), a(0)=1, a(1)=7.
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..n} 6^k*A063967(n,k). - Philippe Deléham, Nov 03 2006
MAPLE
a:= n-> (<<0|1>, <7|7>>^n. <<1, 7>>)[1, 1]:
seq(a(n), n=0..30);
MATHEMATICA
Join[{a=0, b=1}, Table[c=7*b+7*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{7, 7}, {1, 7}, 30] (* Harvey P. Dale, Nov 30 2012 *)
PROG
(Sage) [lucas_number1(n, 7, -7) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009
(PARI) Vec(1/(1-7*x-7*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
(Magma) I:=[1, 7]; [n le 2 select I[n] else 7*Self(n-1) + 7*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018
CROSSREFS
Cf. A000045.
Sequence in context: A092315 A229248 A242159 * A156362 A055274 A152776
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Aug 11 2000
STATUS
approved

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Last modified July 25 12:21 EDT 2024. Contains 374588 sequences. (Running on oeis4.)