|
%I #48 Dec 30 2023 23:52:56
%S 1,9,90,891,8829,87480,866781,8588349,85096170,843160671,8354311569,
%T 82777250160,820184055561,8126651751489,80521522263450,
%U 797833566134451,7905195795581109,78327264255440040,776092140459190341,7689774642431673429,76192801046017773930
%N Scaled Chebyshev U-polynomials evaluated at i*3/2. Generalized Fibonacci sequence.
%C 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.
%C 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
%C a(n) is the number of n-digit integers which do not have two consecutive zeros. - _Ran Pan_, Jan 26 2016
%H Colin Barker, <a href="/A057092/b057092.txt">Table of n, a(n) for n = 0..1000</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=9, q=9.
%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=9.
%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 (9,9).
%F a(n) = 9*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
%F 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.
%F G.f.: 1/(1-9*x-9*x^2).
%F a(n) = Sum_{k, 0<=k<=n}8^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%t Join[{a=0,b=1},Table[c=9*b+9*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)
%t LinearRecurrence[{9,9}, {1,9}, 50] (* _G. C. Greubel_, Jan 25 2018 *)
%o (Sage) [lucas_number1(n,9,-9) for n in range(1, 20)] # _Zerinvary Lajos_, Apr 26 2009
%o (PARI) Vec(1/(1-9*x-9*x^2) + O(x^30)) \\ _Colin Barker_, Jun 14 2015
%o (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
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 11 2000
|
