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%I #49 Dec 30 2023 23:52:42
%S 1,10,110,1200,13100,143000,1561000,17040000,186010000,2030500000,
%T 22165100000,241956000000,2641211000000,28831670000000,
%U 314728810000000,3435604800000000,37503336100000000,409389409000000000,4468927451000000000,48783168600000000000
%N Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized Fibonacci sequence.
%C This is the m=10 member of the m-family of sequences a(m,n)= S(n,i*sqrt(m))*(-i*sqrt(m))^n, with S(n,x) given in Formula and g.f.: 1/(1-m*x-m*x^2). The instances m=1..9 are A000045 (Fibonacci), A002605, A030195, A057087, A057088, A057089, A057090, A057091, A057092.
%C With the roots rp(m) := (m+sqrt(m*(m+4)))/2 and rm(m) := (m-sqrt(m*(m+4)))/2 the Binet form of these m-sequences is a(n,m)= (rp(m)^(n+1)-rm(m)^(n+1))/(rp(m)-rm(m)).
%C a(n) gives the length of the word obtained after n steps with the substitution rule 0->1^10, 1->(1^10)0, starting from 0. The number of 1's and 0's of this word is 10*a(n-1) and 10*a(n-2), resp.
%H Colin Barker, <a href="/A057093/b057093.txt">Table of n, a(n) for n = 0..963</a>
%H Martin Burtscher, Igor Szczyrba, and 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=10, q=10.
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H Wolfdieter 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=10.
%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 (10,10).
%F a(n) = 10*(a(n-1) + a(n-2)), a(-1)=0, a(0)=1.
%F a(n) = S(n, i*sqrt(10))*(-i*sqrt(10))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
%F G.f.: 1/(1 - 10*x - 10*x^2).
%F a(n) = Sum_{k=0..n} 9^k*A063967(n,k). - _Philippe Deléham_, Nov 03 2006
%t Join[{a=0,b=1},Table[c=10*b+10*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011 *)
%o (Sage) [lucas_number1(n,10,-10) for n in range(1, 19)] # _Zerinvary Lajos_, Apr 26 2009
%o (PARI) Vec(1/(1-10*x-10*x^2) + O(x^30)) \\ _Colin Barker_, Jun 14 2015
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Aug 11 2000
%E Extended by _T. D. Noe_, May 23 2011
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