NSW nos

[Ralf Stephan]

> %F A002315 a(n) = 1/2((1+sqrt(2))^(2*n+1) + (1-sqrt(2))^(2*n+1)).
> %F A002315 a(n)=sqrt(2*(A001653(n))^2-1). G.f.: (1+x)/(1-6*x+x^2).
> %F A002315 a(n)= S(n,6)+S(n-1,6) = S(2*n,sqrt(8)), S(n,x)=U(n,x/2) are Chebyshev's polynomials of the 2nd kind. Cf. A049310. S(n,6)= A001109(n+1).
> %F A002315 a(n) ~ 1/2*(sqrt(2) + 1)^(2*n+1) - Joe Keane (jgk at jgk.org), May 15 2002
> %F A002315 Lim n -> inf. a(n)/a(n-1) = 3 + 2*Sqrt(2). - Gregory V. Richardson (omomom at hotmail.com), Oct 06 2002
> %F A002315 a(n) = [2*[(3+2*Sqrt(2))^n - (3-2*Sqrt(2))^n] - 5*[(3+2*Sqrt(2))^(n-1) - (3- 2*Sqrt(2))^(n-1)] + [(3+2*Sqrt(2))^(n-2) - (3-2*Sqrt(2))^(n-2)] ] / (4*Sqrt(2)) - Gregory V. Richardson (omomom at hotmail.com), Oct 06 2002

If you don't like 2n in the exponent, the most elementary 
formula (in the sense of a*b^n+c*d^n) is 

a(n)=(1+sqrt(2))/2*(3+sqrt(8))^n+(1-sqrt(2))/2*(3-sqrt(8))^n.


ralf