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Triangular Triples (Talmadge formula)

Entire pursuit inspired by this James Tanton tweet. Consider the "triangular number function" T(N) = 1+2+3+…+N = N(N+1)/2. The referenced tweet invites the challenge: How many "Triangular Triples" satisfy T(A)+T(B)=T(C) for positive integers A, B, and C, and does there exist a formula or algorithm to generate all of them?