aequationes mathematicae

, Volume 50, Issue 1–2, pp 73–94

# On the representation of integers as sums of triangular numbers

• Ken Ono
• Sinai Robins
• Patrick T. Wahl
Survey Papers

## Summary

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)δ24(n − 3). Furthermore the formula forδ24(n) involves the Ramanujanτ(n)-function. As a consequence, we get elementary congruences forτ(n). In a similar vein, whenp is a prime, we demonstrateδ24(p k − 3) as a Dirichlet convolution ofσ11(n) andτ(n). It is also of interest to know that this study produces formulas for the number of lattice points insidek-dimensional spheres.

## AMS (1991) subject classification

Primary 11F11, 11F12, 11F37 Secondary 11P81

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© Birkhäuser Verlag 1995

## Authors and Affiliations

• Ken Ono
• 1
• Sinai Robins
• 2
• Patrick T. Wahl
• 3
1. 1.Department of MathematicsThe University of GeorgiaAthensUSA
2. 2.Department of Mathematical SciencesUniversity of Northern ColoradoGreeleyUSA
3. 3.Department of MathematicsUniversity of ColoradoBoulderUSA