Continued fractions as inverted recurrence relations

[Joseph Biberstine]

As in http://functions.wolfram.com/Constants/Pi/10/0006/, we see the 
following:

4/Pi = 1 + 1/(3 + 4/(5 + 9/(7 + 16/(9 + 25/(11 + ...))))) that is, this 
is a non-simple cont.frac. where the numerators are the squares and the 
partial quotients are the odds.

Consider a[n] = 2*n - 1 + n^2/a[n+1] with a[1] = 4/Pi.

Solving the above recurrence in the other direction we would have a[n] = 
  (n-1)^2/(a[n-1 - 2*n + 3) with a[1] = 4/Pi.

Now consider this last defined sequence a[n].  We suspect it grows 
linearly, (1) does it?  (2) What is the limit of a[n]/n as n->Infinity? 
  If this limit exists, (3) is it in the OEIS and (4) can it be 
expressed in closed form?  We suspect the value of a[1] is largely 
inconsequential, (5) is it?  If this limit exists, (6) how if at all 
does it relate to Pi?

If the limit does exist, it appear to be approximately 0.414 (roughly 
a[2000]/2000), though I have no means to calculate it further.

- J. Biberstine