%I A057093
%S A057093 1,10,110,1200,13100,143000,1561000,17040000,186010000,2030500000,
%T A057093 22165100000,241956000000,2641211000000,28831670000000,314728810000000,
%U A057093 3435604800000000,37503336100000000,409389409000000000
%N A057093 Scaled Chebyshev U-polynomials evaluated at i*sqrt(10)/2. Generalized 
               Fibonacci sequence.
%C A057093 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, A057087-92.
%C A057093 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 A057093 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.
%D A057093 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=10, q=10.
%D A057093 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs.(39) and 
               (45),rhs, m=10.
%H A057093 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057093 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057093 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057093 a(n) = 10*(a(n-1)+a(n-2)), a(-1)=0, a(0)=1.
%F A057093 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 A057093 G.f.: 1/(1-10*x-10*x^2).
%F A057093 a(n)=Sum_{k, 0<=k<=n}9^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A057093 a(n)=-(1/70)*[5-sqrt(35)]^(n+1)*sqrt(35)+(1/70)*sqrt(35)*[5+sqrt(35)]^(n+1), 
               with n>=0 [From Paolo P. Lava (ppl(AT)spl.at), Nov 20 2008]
%o A057093 (Other) sage: [lucas_number1(n,10,-10) for n in xrange(1, 19)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 26 2009]
%Y A057093 Sequence in context: A144099 A102092 A105279 this_sequence A055276 A143749 
               A049398
%Y A057093 Adjacent sequences: A057090 A057091 A057092 this_sequence A057094 A057095 
               A057096
%K A057093 nonn,easy
%O A057093 0,2
%A A057093 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 11 2000