#### Vol. 1, No. 1, 2008

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An asymptotic for the representation of integers as sums of triangular numbers

### Atanas Atanasov, Rebecca Bellovin, Ivan Loughman-Pawelko, Laura Peskin and Eric Potash

Vol. 1 (2008), No. 1, 111–121
##### Abstract

Motivated by the result of Rankin for representations of integers as sums of squares, we use a decomposition of a modular form into a particular Eisenstein series and a cusp form to show that the number of ways of representing a positive integer $n$ as the sum of $k$ triangular numbers is asymptotically equivalent to the modified divisor function ${\sigma }_{2k-1}^{♯}\left(2n+k\right)$.

##### Keywords
modular form, triangular number, asymptotics
Primary: 11F11