, Volume 50, Issue 1-2, pp 73-94

On the representation of integers as sums of triangular numbers

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In this survey article we discuss the problem of determining the number of representations of an integer as sums of triangular numbers. This study yields several interesting results. Ifn ≥ 0 is a non-negative integer, then thenth triangular number isT n =n(n + 1)/2. Letk be a positive integer. We denote byδ k (n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculateδ k (n). The case wherek = 24 is particularly interesting. It turns out that, ifn ≥ 3 is odd, then the number of points on the 24 dimensional Leech lattice of norm 2n is 212(212 − 1)δ 24(n − 3). Furthermore the formula forδ 24(n) involves the Ramanujanτ(n)-function. As a consequence, we get elementary congruences forτ(n). In a similar vein, whenp is a prime, we demonstrateδ 24(p k − 3) as a Dirichlet convolution ofσ 11(n) andτ(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.