Triangular numbers with triangular squares

[Jack Brennen]

N. J. A. Sloane wrote:
> Jack,   this doesn't seem to be in the OEIS at prsent
> - could you please submit it in the usual way?
> Many thanks!
> Neil


> s[0] = 1
> s[1] = 9
> s[n] = 6*s[n-1] - s[n-2] - 4
> 
> The sequence goes:
> 
>   1, 9, 49, 281, 1633, 9513, 55441, 323129, ...

Like Franklin T. Adams-Watters wrote earlier today,
this sequence is simply A001109, put through the
transform 8*x+1.  The most straightforward test
for "triangularity":

    istriangle(n)  <===>  issquare(8*n+1)

So my sequence wouldn't seem to have significant
interest on its own, it's just a sequence which
appears as an intermediate step during some other
work.

But then again, if the sequence were added, and
if a comment was attached stating that if this
sequence could be proven to be free of squares
beyond the first three terms, that would lead
directly to a proof that 0, 1, and 6 are the
only triangular numbers whose squares are
triangular numbers, that could be useful.

Is that sufficient reason to create a new derived
sequence?  That an easy-to-understand conjecture
about the sequence would lead to a desirable
result?

      Jack