A057091 Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence.

1, 8, 72, 640, 5696, 50688, 451072, 4014080, 35721216, 317882368, 2828828672, 25173688320, 224020135936, 1993550594048, 17740565839872, 157872931471360, 1404907978489856, 12502247279689728, 111257242065436672, 990075914761011200, 8810665254611582976

Offset: 0

Author

Wolfdieter Lang, Aug 11 2000

Keywords

• nonn
• easy

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• 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.
• 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=8.
• Index entries for sequences related to Chebyshev polynomials.
• Index entries for linear recurrences with constant coefficients, signature (8,8).

Programs

Magma

I:=[1,8]; [n le 2 select I[n] else 8*Self(n-1) + 8*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 24 2018

Mathematica

LinearRecurrence[{8,8}, {1,8}, 50] (* G. C. Greubel, Jan 24 2018 *)

PARI

Vec(1/(1-8*x-8*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

Sage

[lucas_number1(n,8,-8) for n in range(0, 20)] # Zerinvary Lajos, Apr 25 2009

Formula

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..n} 7^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

a(n) = 2^n*A090017(n+1). - R. J. Mathar, Mar 08 2021