On the representation of integers as sums of triangular numbers

Abstract

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. If n ≥0 is a non-negative integer, then the nth triangular number is T n = n(n + 1)/2. Let k be a positive integer. We denote by δ k (n) the number of representations of n as a sum of k triangular numbers. Here we use the theory of modular forms to calculate δ k (n). The case where k = 24 is particularly interesting. It turns out that, if n ≥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 for δ 24(n) involves the Ramanujan τ(n)-function. As a consequence, we get elementary congruences for τ(n). In a similar vein, when p is a prime, we demonstrate δ 24 (p k − 3) as a Dirichlet convolution of σ 11(n) and τ (n). It is also of interest to know that this study produces formulas for the number of lattice points inside k-dimensional spheres.