For today's foray into Waring theory ... Consider, please, OEIS sequence A002636, the number of ways of expressing n as the sum of 3 triangular numbers. (For the present purposes, 0 counts as a triangular number.) By inspecting the list, I conjecture that 53 is the largest number to have only one tri-triangular representation: 53 = 28 + 15 + 10. Is this very very hard to prove? The largest number I could find with only two tri-triangular representations is 194. The entries (53,194) are enough to see easily that this sequence -- if it is well-defined -- is not in OEIS. I conjecture that for every n in N1, there is a largest k such that A002636(k) = n. How hard is *this* to prove? It looks like when n = 3, k = 470; when n = 4, k = 788; and when n = 5, k = 1730.