Continued Fractions With Square Numerators

[Jack Brennen]

The next few terms of Leroy's sequence are:

1,3,2,5, 43, 522, 1104509, ...

The continued fraction [1,3,2,5,43,522,1104509] corresponds to:

  1220041748025/946166591401

Where the numerator is 1104555 squared.

I'm sure it's not a coincidence that the square root of the numerator
is numerically very close to the last term of the continued fraction,
but I'm not sure yet what that says about the growth of the sequence
or about whether it's infinite or not.

  Jack