TITLE: Formula for $\binom{n}{2}/\binom{m}{2}=1/2$ or $2n(n-1) = m(m-1)$?
QUESTION [6 upvotes]: Where could I find a formula that produces integers $n$ and $m$ such that $\binom{n}{2}/\binom{m}{2}=1/2$?
Of course, this questions can be reformulated as: How to find all the integer values of n and m such that $$2n(n-1) = m(m-1)\quad ?$$

REPLY [4 votes]: In my answer here you will find Lagrange's method for reducing the general bivariate quadratic Diophantine equation to a Pell equation, along with some interesting links, including a Java applet that will show you the complete solution, including complete step-by-step details of the solution.
According to said applet the solutions are given by
$$\rm N_{i+1}\ =\ 3\ N_i + 2\ M_i -2 $$
$$\rm M_{i+1}\ =\ 4\ N_i + 3\ M_i -3 $$
with $\rm\ N_0,M_0\ =\ 0,1\:;\ \ 1,1\:.\ $  To see complete details of the solution, including the relationship with the continued fraction of $\rm\:\sqrt{2}\:,\:$ choose the "step by step" solution method.