How many Pythagorean triples $(a,b,c)$ are there such that $a, b$ and $c$ are triangular?

Any two solutions with only $a$ and $b$ interchanged are considered equivalent.

The question of existence was posed by Zarankiewicz and answered by Sierpinski in *Sur les nombres triangulaires carrés* (1961):

Such a triangle is known whose sides are $t_{132} = 8778, t_{143} = 10296$ and $t_{164} = 13530$, but we do not know if there are others and if their number is finite.

A partial result: Since $t_n^2 = 1^3 + \ldots + n^3$, the equation $t_x^2 + t_y^2 = t_z^2$ is equivalent to $t_x^2$ being a sum of $k = z - y$ consecutive cubes. This leads one to consider elliptic curves.
In *Pythagorean triples and triangular numbers* (1979) by D. W. Ballew and R. C. Weger it is shown by applying a theorem of Siegel that for a given $k$ there are only finitely many solutions. Actually finding such solutions has been done by R. J. Stroeker: *On the sum of consecutive cubes being a perfect square* (1995), without the restriction that the square is some $t_x^2$.