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A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.

%I A057089 #55 Jan 05 2025 19:51:36
%S A057089 1,6,42,288,1980,13608,93528,642816,4418064,30365280,208700064,
%T A057089 1434392064,9858552768,67757668992,465697330560,3200729997312,
%U A057089 21998563967232,151195763787264,1039165966526976,7142170381885440
%N A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence.
%C A057089 a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^6, 1->(1^6)0, starting from 0. The number of 1's and 0's of this word is 6*a(n-1) and 6*a(n-2), resp.
%H A057089 Vincenzo Librandi, <a href="/A057089/b057089.txt">Table of n, a(n) for n = 0..200</a>
%H A057089 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 A057089 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=6, q=6.
%H A057089 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A057089 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=6.
%H A057089 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H A057089 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,6).
%F A057089 a(n) = 6*a(n-1) + 6*a(n-2); a(0)=1, a(1)=6.
%F A057089 a(n) = S(n, i*sqrt(6))*(-i*sqrt(6))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
%F A057089 G.f.: 1/(1-6*x-6*x^2).
%F A057089 a(n) = Sum_{k=0..n} 5^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%t A057089 Join[{a=0,b=1},Table[c=6*b+6*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)
%t A057089 LinearRecurrence[{6,6},{1,6},40] (* _Harvey P. Dale_, Nov 05 2011 *)
%o A057089 (Sage) [lucas_number1(n,6,-6) for n in range(1, 21)] # _Zerinvary Lajos_, Apr 24 2009
%o A057089 (Magma) I:=[1,6]; [n le 2 select I[n] else 6*Self(n-1)+6*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 14 2011
%o A057089 (PARI) x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ _G. C. Greubel_, Jan 24 2018
%Y A057089 Cf. A001076, A006190, A007482, A015520, A015521, A015523, A015524, A015525, A015528, A015529, A015530, A015531, A015532, A015533, A015534, A015535, A015536, A015537, A015440, A015441, A015443, A015444, A015445, A015447, A015548, A030195, A053404, A057087, A057088, A083858, A085939, A090017, A091914, A099012, A135030, A135032, A180222, A180226, A180250.
%K A057089 nonn,easy
%O A057089 0,2
%A A057089 _Wolfdieter Lang_, Aug 11 2000