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%I #47 Dec 30 2023 23:51:14
%S 1,8,72,640,5696,50688,451072,4014080,35721216,317882368,2828828672,
%T 25173688320,224020135936,1993550594048,17740565839872,
%U 157872931471360,1404907978489856,12502247279689728,111257242065436672,990075914761011200,8810665254611582976
%N Scaled Chebyshev U-polynomials evaluated at i*sqrt(2). Generalized Fibonacci sequence.
%C 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.
%H Colin Barker, <a href="/A057091/b057091.txt">Table of n, a(n) for n = 0..1000</a>
%H A. F. Horadam, <a href="http://www.fq.math.ca/Scanned/5-5/horadam.pdf">Special properties of the sequence W_n(a,b; p,q)</a>, Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=8.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Wolfdieter Lang, <a href="http://www.fq.math.ca/Scanned/38-5/lang.pdf">On polynomials related to powers of the generating function of Catalan's numbers</a>, Fib. Quart. 38 (2000) 408-419. Eqs.(39) and (45),rhs, m=8.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,8).
%F a(n) = 8*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
%F 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.
%F G.f.: 1/(1 - 8*x - 8*x^2).
%F a(n) = Sum_{k=0..n} 7^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%F a(n) = 2^n*A090017(n+1). - _R. J. Mathar_, Mar 08 2021
%t LinearRecurrence[{8,8}, {1,8}, 50] (* _G. C. Greubel_, Jan 24 2018 *)
%o (Sage) [lucas_number1(n,8,-8) for n in range(0, 20)] # _Zerinvary Lajos_, Apr 25 2009
%o (PARI) Vec(1/(1-8*x-8*x^2) + O(x^30)) \\ _Colin Barker_, Jun 14 2015
%o (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
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 11 2000
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