A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence.

1, 7, 56, 441, 3479, 27440, 216433, 1707111, 13464808, 106203433, 837677687, 6607167840, 52113918689, 411047605703, 3242130670744, 25572247935129, 201700650241111, 1590910287233680, 12548276562323537, 98974307946900519, 780658091564568392

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

Colin Barker, Table of n, a(n) for n = 0..1000

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.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (7,7).

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

Adjacent sequences: A057087 A057088 A057089 * A057091 A057092 A057093

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 11 2000

Status

approved