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A057091 Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence. 11
1, 8, 72, 640, 5696, 50688, 451072, 4014080, 35721216, 317882368, 2828828672, 25173688320, 224020135936, 1993550594048, 17740565839872, 157872931471360, 1404907978489856, 12502247279689728, 111257242065436672, 990075914761011200, 8810665254611582976 (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->1^8, 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

W. 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).

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<=k<=n}7^k*A063967(n,k). - Philippe Deléham, 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, Jul 08 2008

MATHEMATICA

Join[{a=0, b=1}, Table[c=8*b+8*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)

PROG

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

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

CROSSREFS

Sequence in context: A229249 A242160 A062541 * A156566 A055275 A155198

Adjacent sequences:  A057088 A057089 A057090 * A057092 A057093 A057094

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang Aug 11 2000

STATUS

approved

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Last modified November 20 00:42 EST 2017. Contains 294957 sequences.