TITLE: For what values of $n$ is $n(n+2)$ a triangle number?
QUESTION [7 upvotes]: A triangle number is a number which can be written in the form $\frac{m(m+1)}{2}$ for some natural number $m$. For what values of $n$ is $n(n+2)$ a triangle number ? 
Using a a brute force method I could generate first $24$ such values of $n$ , but beyond that its to slow to generate. Here are the first $24$ values of $n$ such that $n(n+2)$ is a triangle number:
1
3
10
22
63
133
372
780
2173 
4551
12670
26530
73851
154633
430440
901272
2508793
5253003
14622322
30616750
85225143
178447501
496728540
1040068260

PS: This is the final step for question swapping counters. I would appreciate hints only.

REPLY [2 votes]: Rather than looking at a direct solution to find $n$ values, you can look for a solution to find m values and using which you can find the values of $n$ respectively.
The m values can be seen to follow a dynamic programming pattern, if you look for it. For help, you can look at oeis link (A006451).
To get the values of $n$ from the $m$ values, we can simply use the formulae for solving the quadratic equation and write n as 
$$
n = \sqrt{\bigg(\frac{y^2 + y +2}2\bigg)} - 1 
$$