Triangular numbers with triangular squares

[Jack Brennen]

I'm pretty sure that 0, 1, and 6 are the only triangular numbers
with a triangular square.

If I'm not mistaken, all solutions can be isolated to this
recursive sequence:

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, ...

If a number X of this sequence is a perfect square, then (X-1)/8
is a triangular number.  Note that due to the form of the
sequence, ((X-1)/8)^2 is always a triangular number.

This sequence grows very quickly and did not produce any more
squares even searching up to 10000 digit numbers.

  Jack