aequationes mathematicae

, Volume 50, Issue 1–2, pp 73–94 | Cite as

On the representation of integers as sums of triangular numbers

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


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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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

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