%I A057080
%S A057080 1,9,71,559,4401,34649,272791,2147679,16908641,133121449,1048062951,
%T A057080 8251382159,64962994321,511452572409,4026657584951,31701808107199,
%U A057080 249587807272641,1965000650073929,15470417393318791,121798338496476399
%N A057080 Even-indexed Chebyshev U-polynomials evaluated at sqrt(10)/2.
%C A057080 a(n) = L(n,-8)*(-1)^n, where L is defined as in A108299; see also A070997 
               for L(n,+8). - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 01 2005
%C A057080 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 A057080 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=10.
%H A057080 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A057080 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A057080 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A057080 For all elements x of the sequence, 15*x^2 + 10 is a square. Lim. n-> 
               Inf. a(n)/a(n-1) = 4 + sqrt(15). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A057080 a(n) = 8*a(n-1)-a(n-2), a(-1)=-1, a(0)=1.
%F A057080 a(n) = S(n, 8)+S(n-1, 8) = S(2*n, sqrt(10)) with S(n, x) := U(n, x/2), 
               Chebyshev polynomials of 2nd kind, A049310. S(n, 8) = A001090(n).
%F A057080 G.f.: (1+x)/(1-8*x+x^2).
%F A057080 a(n) = [ [(4+sqrt(15))^(n+1) - (4-sqrt(15))^(n+1)] + [(4+sqrt(15))^n 
               - (4-sqrt(15))^n] ] / (2*sqrt(15)) - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 13 2002
%F A057080 Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, 
               -10)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002
%F A057080 a(n)=Jacobi_P(n,1/2,-1/2,4)/Jacobi_P(n,-1/2,1/2,1); - Paul Barry (pbarry(AT)wit.ie), 
               Feb 03 2006
%F A057080 a(n+1)=4*a(n)+((3*a(n)^2+2)*5)^0.5. - Richard Choulet (richardchoulet(AT)yahoo.fr), 
               Aug 30 2007
%o A057080 (Other) sage: [(lucas_number2(n,8,1)-lucas_number2(n-1,8,1))/6 for n 
               in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Nov 10 2009]
%Y A057080 A033890. a(n)=sqrt((5*A070997(n)^2 - 2)/3) (cf. Richardson comment).
%Y A057080 Sequence in context: A156705 A081900 A164551 this_sequence A001706 A158193 
               A123987
%Y A057080 Adjacent sequences: A057077 A057078 A057079 this_sequence A057081 A057082 
               A057083
%K A057080 nonn
%O A057080 0,2
%A A057080 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Aug 04 2000