The triangular numbers are the integers m(m + 1)/2, m = 0, 1, 2, . . . . For a positive integer k, we let δ k(n) denote the number of representations of the nonnegative integer n as the sum of k triangular numbers. In 1994, using advanced methods, Kac and Wakimoto gave formulae for δ 16(n) and δ 24(n). Using a recent elementary identity due to Huard, Ou, Spearman and Williams, elementary proofs are given of these formulae. Copyright

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Journal Rocky Mountain Journal of Mathematics
Huard, J.G. (James G.), & Williams, K.S. (2005). Sums of sixteen and twenty-four triangular numbers. Rocky Mountain Journal of Mathematics, 35(3), 857–868.