Continued Fractions With Square Numerators

[Leroy Quet]

Define an integer sequence as follows:

a(1) = 1.

a(n) is the lowest positive integer such that the continued fraction
[a(1);a(2),a(3),...,a(n)]
is equal to a rational with a square numerator.

The sequence begins:

1, 3, 2, 5,...

So, for example, to get the fourth term, we have the continued fraction
[1; 3, 2, m],
and we want to find the lowest positive integer m such that the ratio
has a square numerator.
[1; 3, 2, 5] = 49/38.

Is this sequence in the EIS?
(http://www.research.att.com/~njas/sequences/index.html#L)
Could someone please calculate/submit its terms if it is not yet in the 
EIS.


(And I am just assuming it is an infinite sequence. Is it?)

And what, I wonder, is the infinite continued fraction (almost) equal to 
numerically?

thanks,
Leroy Quet