Continued fractions as inverted recurrence relations

[Paul D. Hanna]

Joseph and Seqfans,
     It appears that the ROUNDed values of the sequence you describe is
in the OEIS, 
but unexpectedly as: 
A086377: a(1)=1; a(n)=a(n-1)+2 if n is in the sequence; a(n)=a(n-1)+2 if
n and (n-1) are not in the sequence; a(n)=a(n-1)+3 if n is not in the
sequence but (n-1) is in the sequence.
 
BUT, THERE IS AN ERROR in A086377 at position 26 and 27:
instead of A086377(26)=6 and A086377(27)=2, it should be joined together
as A086377(26)=62.
 
So, given Joseph Bibers definition of b(n): 
b[n] = (n-1)^2/(b[n-1] - 2*n + 3) with b[1] = 4/Pi,
then 
round( b(n) ) = A086377(n).
 
Does this equivalence continue to hold?  If so, why?
    Paul
 
> 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?
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