[seqfan] Minimal k > n such that (4k+3n)(4n+3k) is square

[Charles Greathouse]

I noticed that a(n) < 393*a(n) for
http://oeis.org/A083752
and wondered if this could be sharpened. It seems that for many n, a(n) =
393n, but if not then a(n) <= 27.25n, with many n (obviously multiples of
4) having this precise value.

The graph of A083752/A27 is compelling. The following conjecture is
numerically natural, though I have no theoretical reason to believe it yet:

If there is no n < k < 109n/4 with (4k+3n)(4n+3k) square, then a(n) = 393n.

Can someone find either a counterexample or a proof?

I have a feeling that the method of a proof might lead to a more efficient
algorithm for the sequence (aside from the obvious optimization of skipping
the associated k).

Charles Greathouse
Analyst/Programmer
Case Western Reserve University