A057092 Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci sequence.

1, 9, 90, 891, 8829, 87480, 866781, 8588349, 85096170, 843160671, 8354311569, 82777250160, 820184055561, 8126651751489, 80521522263450, 797833566134451, 7905195795581109, 78327264255440040, 776092140459190341, 7689774642431673429, 76192801046017773930

Offset

0,2

Comments

a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^9, 1->(1^9)0, starting from 0. The number of 1's and 0's of this word is 9*a(n-1) and 9*a(n-2), resp.

a(n) gives the number of n-digit integers which have no digit repeated 3 times in a row. Example: a(2)= 90 which is all the 2-digit integers. a(3) = 891 = all 900 3-digit integers except 111, 222, 333, ..., 999. - Toby Gottfried, Apr 01 2013

a(n) is the number of n-digit integers which do not have two consecutive zeros. - Ran Pan, Jan 26 2016

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Martin Burtscher, Igor Szczyrba, 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=9, q=9.

Tanya Khovanova, Recursive Sequences

W. 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=9.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (9,9).

Formula

a(n) = 9*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.

a(n) = S(n, i*3)*(-i*3)^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.

G.f.: 1/(1-9*x-9*x^2).

a(n) = Sum_{k, 0<=k<=n}8^k*A063967(n,k). - Philippe Deléham, Nov 03 2006

Mathematica

Join[{a=0, b=1}, Table[c=9*b+9*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011 *)
LinearRecurrence[{9, 9}, {1, 9}, 50] (* G. C. Greubel, Jan 25 2018 *)

Prog

(Sage) [lucas_number1(n, 9, -9) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009
(PARI) Vec(1/(1-9*x-9*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015
(Magma) I:=[1, 9]; [n le 2 select I[n] else 9*Self(n-1) + 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 25 2018

Crossrefs

Sequence in context: A242161 A227713 A343366 * A156577 A299872 A173480

Adjacent sequences: A057089 A057090 A057091 * A057093 A057094 A057095

Keyword

nonn,easy

Author

Wolfdieter Lang, Aug 11 2000

Status

approved