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

Author

Wolfdieter Lang, Aug 11 2000

Keywords

• nonn
• easy

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).

Programs

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

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 *)

PARI

Vec(1/(1-9*x-9*x^2) + O(x^30)) \\ Colin Barker, Jun 14 2015

Sage

[lucas_number1(n,9,-9) for n in range(1, 20)] # Zerinvary Lajos, Apr 26 2009

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