TITLE: Determine if a number is the sum of two triangular numbers.
QUESTION [13 upvotes]: Is it possible to figure out if a number $z$ is the addition of two triangular number without recursion or finding the values to $x$ and $y$?
$$\frac{x(x+1)}{2} + \frac{y(y+1)}{2} = z$$
An example of this would be following: 
$$\frac{345(345+1)}{2} + \frac{234(234+1)}{2} = 87180$$
Is it possible to determine if $87180$ originated from the addition of two triangular numbers without recursively going back and plugging every possibility for $x$ and $y$?

REPLY [2 votes]: If we had $a(k)$ from A008441 the test for $z$ would be
$$
a(z) > 0 .
$$
The information section on this series provides many interesting relations.