From: Max Alekseyev (maxale(AT)gmail.com)
Date: Jul 31, 2007 6:06 PM

Notes on A024537.

I will show that the explicit formula for A024537:
(*)   a(n) = 1/4*( 2 + (1-Sqrt(2))^(n+1) + (1+Sqrt(2))^(n+1) )
satisfies the recurrence relation 
a(n+1) = [a(n)^2 / a(n-1)] + 1
and below we will show that this formula follows from (*).

To simplify notation, let c=1+Sqrt(2).
Using formula (*), it is easy to get that:
a(n)^2 / a(n-1) = ( c^(n+2) + 4*c - 2*c^2 ) / 4 + O(c^(-n))
= ( c^(n+2) - 2 ) / 4 + O(c^(-n)) = a(n+1) - 1 + O(c^(-n))
where the terms hidden inside O() are positive and quickly become
negligible as n grows.

Therefore [a(n)^2 / a(n-1)] + 1 = a(n+1).
Q.E.D.