Continued Fractions With Square Numerators

[jens at voss-ahrensburg.de]

> 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,

Yes, it is indeed not a coincidence; see below.


> but I'm not sure yet what that says about the growth of the sequence
> or about whether it's infinite or not.

Well, here's a different way of looking at Leroy's sequence:

Let r_0 := 0 and r_1 := 1. Then for all positive integers i, Leroy's
a_i is simply the smallest positive integer for which
r_(i+1) := r_i * a_i + r_(i-1) is a square.

Since r_i * [r_i - 2 * sqrt(r_(i-1))] + r_(i-1) is always a square (and
r_i * [r_i / 4 - sqrt(r_(i-1))] + r_(i-1) is a square for even r_i)
a_i is bounded by r_i - 2 * sqrt(r_(i-1)) (resp. r_i / 4 -
sqrt(r_(i-1)));
therefore the sequence is infinite.

Regards,
Jens
--
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