aequationes mathematicae

, Volume 50, Issue 1, pp 73–94

On the representation of integers as sums of triangular numbers

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

DOI: 10.1007/BF01831114

Cite this article as:
Ono, K., Robins, S. & Wahl, P.T. Aeq. Math. (1995) 50: 73. doi:10.1007/BF01831114


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 isTn =n(n + 1)/2. Letk be a positive integer. We denote byδk(n) the number of representations ofn as a sum ofk triangular numbers. Here we use the theory of modular forms to calculateδk(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(pk − 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, 11F37Secondary 11P81

Copyright information

© 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