Continued fractions as inverted recurrence relations

[Joseph Biberstine]

Paul D. Hanna wrote:
 > A086377
Ah, noted; thank you for that observation.


Don Reble wrote:
a[n]/n appears to converge to -0.41421...
 >     Alas, that doesn't explain Joseph's +0.414.
Yes, apologies for this indiscretion.  I modified the code to test 
another thing and forgot this detail.


Paul D. Hanna wrote:
 > round( b(n) ) = floor((1+sqrt(2))*n-1/sqrt(2)) = A086377(n).
 >
 > I get that the first 190 terms agree - can someone check further?
 > And perhaps supply some theory as to why?
 >       Paul
I would also like to see this.


 >  From the recurrence relation above, it follows that the L=lim a[n]/n
 > satisfies the following quadratic equation: L^2 - 2*L - 1 = 0 meaning
 > that L = 1+sqrt(2) or 1-sqrt(2).
 >
 > Max
 >
Oh, of course, I'm surprised I forgot that notion.  Thanks.