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

%I A057090 #53 Jan 05 2025 19:51:36
%S A057090 1,7,56,441,3479,27440,216433,1707111,13464808,106203433,837677687,
%T A057090 6607167840,52113918689,411047605703,3242130670744,25572247935129,
%U A057090 201700650241111,1590910287233680,12548276562323537,98974307946900519,780658091564568392
%N A057090 Scaled Chebyshev U-polynomials evaluated at i*sqrt(7)/2. Generalized Fibonacci sequence.
%C A057090 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.
%H A057090 Colin Barker, <a href="/A057090/b057090.txt">Table of n, a(n) for n = 0..1000</a>
%H A057090 Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A057090 A. F. Horadam, <a href="https://web.archive.org/web/2024*/https://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=7, q=7.
%H A057090 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A057090 Wolfdieter Lang, <a href="https://web.archive.org/web/2024*/https://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=7.
%H A057090 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A057090 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,7).
%F A057090 a(n) = 7*(a(n-1) + a(n-2)), a(0)=1, a(1)=7.
%F A057090 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.
%F A057090 G.f.: 1/(1 - 7*x - 7*x^2).
%F A057090 a(n) = Sum_{k=0..n} 6^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%p A057090 a:= n-> (<<0|1>, <7|7>>^n. <<1, 7>>)[1, 1]:
%p A057090 seq(a(n), n=0..30);
%t A057090 Join[{a=0,b=1},Table[c=7*b+7*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)
%t A057090 LinearRecurrence[{7,7},{1,7},30] (* _Harvey P. Dale_, Nov 30 2012 *)
%o A057090 (Sage) [lucas_number1(n,7,-7) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 24 2009
%o A057090 (PARI) Vec(1/(1-7*x-7*x^2) + O(x^30)) \\ _Colin Barker_, Jun 14 2015
%o A057090 (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
%Y A057090 Cf. A000045.
%K A057090 nonn,easy
%O A057090 0,2
%A A057090 _Wolfdieter Lang_, Aug 11 2000