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 A057089 Scaled Chebyshev U-polynomials evaluated at i*sqrt(6)/2. Generalized Fibonacci sequence. 14
 1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440 (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->1^6, 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. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Martin Burtscher, Igor Szczyrba, and Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5. 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=6, q=6. Tanya Khovanova, Recursive Sequences Wolfdieter 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=6. Index entries for linear recurrences with constant coefficients, signature (6,6). FORMULA a(n) = 6*a(n-1) + 6*a(n-2); a(0)=1, a(1)=6. 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. G.f.: 1/(1-6*x-6*x^2). a(n) = Sum_{k=0..n} 5^k*A063967(n,k). - Philippe Deléham, Nov 03 2006 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 MATHEMATICA Join[{a=0, b=1}, Table[c=6*b+6*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *) LinearRecurrence[{6, 6}, {1, 6}, 40] (* Harvey P. Dale, Nov 05 2011 *) PROG (Sage) [lucas_number1(n, 6, -6) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009 (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 (PARI) x='x+O('x^30); Vec(1/(1-6*x-6*x^2)) \\ G. C. Greubel, Jan 24 2018 CROSSREFS 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. Sequence in context: A105482 A242158 A157335 * A110711 A156361 A216517 Adjacent sequences: A057086 A057087 A057088 * A057090 A057091 A057092 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Aug 11 2000 STATUS approved

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Last modified December 5 04:50 EST 2022. Contains 358578 sequences. (Running on oeis4.)