Allan, You should look at this paper by Ono, Robbins and Wahl http://www.mathcs.emory.edu/~ono/publications-cv/pdfs/006.pdf . In it, among other things, they give an explicit formula for the number of representations of n as a sum of 3 triangular numbers. Victor On Wed, Jan 15, 2014 at 2:13 PM, Allan Wechsler <acwacw@gmail.com> wrote:
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. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun