[math-fun] sum of 2 triangular numbers is a triangular number

Let T(i) = i(i+1)/2. Given n, let k be smallest number such that T(n) + T(k) = T(m) for some m. The k and m values are in A082183 and A082184. It must be classical that k and m always exist. - can someone supply a reference or a proof? The graph of the k values is quite irregular. Is there an upper bound?