Sums Of Triangular Numbers Research Articles

Pythagorean triples and sums of triangular numbers

Each primitive Pythagorean triple of positive integers leads, with the aid of an algebraic identity, to a family of triples of integers. Each triple in this family provides three triangular numbers, one for each component, such that the sum of two of them is equal to the third one.

Proof without Words: Alternating Sums of Triangular Numbers

(1995). Proof without Words: Alternating Sums of Triangular Numbers. Mathematics Magazine: Vol. 68, No. 4, pp. 284-305.

On the representation of integers as sums of triangular numbers

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.

On Sums of Triangular Numbers and Sums of Squares

(1992). On Sums of Triangular Numbers and Sums of Squares. The American Mathematical Monthly: Vol. 99, No. 8, pp. 752-757.

On Sums of Triangular Numbers and Sums of Squares

On Sums of Triangular Numbers and Sums of Squares

Proof without Words: Sums of Triangular Numbers

Proof without Words: Sums of Triangular Numbers

Proof without Words: Sums of Triangular Numbers

Proof without Words: Sums of Triangular Numbers

2939. On the representation of numbers as sums of triangular numbers

2939. On the representation of numbers as sums of triangular numbers