0,2

a(n) = S(n,7)+S(n-1,7) = S(2*n,sqrt(9) = 3), S(n,x) = U(n,x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n,7) = A004187(n+1), S(n,3) = A001906(n+1).

(x,y)=(a(n),a(n+1)) are solutions of (x+y)^2/(1+xy)=9, the other solutions are in A033888. - Floor van Lamoen (fvlamoen(AT)hotmail.com), Dec 10 2001

The sequence A033890 provides half of the solutions to the equation 5*x^2 + 4 is a square. The other solutions are included in A033888. Lim. n-> Inf. a(n)/a(n-1) = phi^4 = (7 + 3*Sqrt(5))/2 - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

a(n) = L(n,-7)*(-1)^n, where L is defined as in A108299; see also A049685 for L(n,+7). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Jun 01 2005

Tanya Khovanova, Recursive Sequences

Index entries for sequences related to Chebyshev polynomials.

G.f.: (1+x)/(1-7*x+x^2). a(n)=7*a(n-1)-a(n-2), n>1. a(0)=1, a(1)=8.

a(n) = [ [(7+3*Sqrt(5))^n - [(7-3*Sqrt(5))^n] + 2*[(7+3*Sqrt(5))^(n-1) - [(7-3*Sqrt(5))^(n-1)] ] / (3*(2^n)*Sqrt(5)) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 13 2002

Let q(n, x)=sum(i=0, n, x^(n-i)*binomial(2*n-i, i)); then (-1)^n*q(n, -9)=a(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Nov 10 2002

Define f[x,s] = s x + Sqrt[(s^2-1)x^2+1]; f[0,s]=0. a(n) = f[a(n-1),7/2] + f[a(n-2),7/2]. - Marcos Carreira, Dec 27 2006

(PARI) a(n)=fibonacci(4*n+2)

Adjacent sequences: A033887 A033888 A033889 this_sequence A033891 A033892 A033893

Sequence in context: A110184 A013698 A075734 this_sequence A010924 A010918 A019484

nonn

njas