%I A057081
%S A057081 1,10,89,791,7030,62479,555281,4935050,43860169,389806471,3464398070,
%T A057081 30789776159,273643587361,2432002510090,21614379003449,192097408520951,
%U A057081 1707262297685110,15173263270645039,134852107138120241
%N A057081 Even indexed Chebyshev U-polynomials evaluated at sqrt(11)/2.
%C A057081 This is the m=11 member of the m-family of sequences S(n,m-2)+S(n-1,m-2) 
               = S(2*n,sqrt(m)) (for S(n,x) see Formula). The m=4..10 instances 
               are: A005408, A002878, A001834, A030221, A002315, A033890 and A057080, 
               resp. The m=1..3 (signed) sequences are: A057078, A057077 and A057079, 
               resp.
%C A057081 a(n) = L(n,-9)*(-1)^n, where L is defined as in A108299; see also A070998 
               for L(n,+9). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 01 2005
%C A057081 General recurrence is a(n)=(a(1)-1)*a(n-1)-a(n-2), a(1)>=4, lim n->infinity 
               a(n)= x*(k*x+1)^n, k =(a(1)-3), x=(1+sqrt((a(1)+1)/(a(1)-3)))/2. 
               Examples in OEIS: a(1)=4 gives A002878, primes in it A121534. a(1)=5 
               gives A001834, primes in it A086386. a(1)=6 gives A030221, primes 
               in it not in OEIS {29,139,3191,...}. a(1)=7 gives A002315, primes 
               in it A088165. a(1)=8 gives A033890, primes in it not in OEIS (does 
               there exist any ?). a(1)=9 gives A057080, primes in it not in OEIS 
               {71,34649,16908641,...}. a(1)=10 gives A057081, primes in it not 
               in OEIS {389806471,192097408520951,...}. [From Ctibor O. Zizka (ctibor.zizka(AT)seznam.cz), 
               Sep 02 2008]
%D A057081 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eq.(44), rhs, 
               m=11.
%H A057081 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057081 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057081 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057081 a(n) = 9*a(n-1)-a(n-2), a(-1)=-1, a(0)=1.
%F A057081 a(n)= S(n, 9)+S(n-1, 9)= S(2*n, sqrt(11)) with S(n, x) := U(n, x/2), 
               Chebyshev polynomials of 2nd kind, A049310. S(n, 9)= A018913(n).
%F A057081 G.f.: (1+x)/(1-9*x+x^2).
%F A057081 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, 
               -11)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002
%o A057081 (Other) sage: [(lucas_number2(n,9,1)-lucas_number2(n-1,9,1))/7 for n 
               in xrange(1, 20)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 10 2009]
%Y A057081 Sequence in context: A000826 A031416 A120923 this_sequence A024132 A044261 
               A065690
%Y A057081 Adjacent sequences: A057078 A057079 A057080 this_sequence A057082 A057083 
               A057084
%K A057081 nonn,new
%O A057081 0,2
%A A057081 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 04 2000