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A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence. 13


%S 1,6,42,288,1980,13608,93528,642816,4418064,30365280,208700064,

%T 1434392064,9858552768,67757668992,465697330560,3200729997312,

%U 21998563967232,151195763787264,1039165966526976,7142170381885440

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

%C 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 Vincenzo Librandi, <a href="/A057089/b057089.txt">Table of n, a(n) for n = 0..200</a>

%H Martin Burtscher, Igor Szczyrba, 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 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=6, q=6.

%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

%H W. 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=6.

%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 (6,6).

%F a(n) = 6*a(n-1)+6*a(n-2), a(0)=1, a(1)=6

%F 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 G.f.: 1/(1-6*x-6*x^2).

%F a(n) = Sum_{k, 0<=k<=n}5^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006

%F a(n) = -(1/30)*sqrt(15)*[3-sqrt(15)]^(n+1)+(1/30)*sqrt(15)*[3+sqrt(15)]^(n+1), with n>=0. [_Paolo P. Lava_, Nov 20 2008]

%t 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 LinearRecurrence[{6,6},{1,6},40] (* _Harvey P. Dale_, Nov 05 2011 *)

%o (Sage) [lucas_number1(n,6,-6) for n in xrange(1, 21)] # _Zerinvary Lajos_, Apr 24 2009

%o (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 (PARI) x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ _G. C. Greubel_, Jan 24 2018

%Y 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 nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 11 2000

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Last modified November 12 16:45 EST 2019. Contains 329058 sequences. (Running on oeis4.)