Triangular numbers with triangular squares

[N. J. A. Sloane]

Jack, you said:

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?


Me:  yes, certainly - please send it in!

Thanks

Neil