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

1, 6, 42, 288, 1980, 13608, 93528, 642816, 4418064, 30365280, 208700064, 1434392064, 9858552768, 67757668992, 465697330560, 3200729997312, 21998563967232, 151195763787264, 1039165966526976, 7142170381885440

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 sequences related to Chebyshev polynomials.

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

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