Sums of sixteen and twenty-four triangular numbers

The triangular numbers are the integers m(m + 1)/2, m, = 0, 1, 2, .... For a positive integer k, we let delta(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 delta(16)(n) and delta(24)(n). Using a recent elementary identity due to Huard, On, Spearman and Williams, elementary proofs are given of these formulae.