# Interesting identities involving weighted representations of integers as sums of arbitrarily many squares

## TL;DR

This paper derives formulas for counting weighted representations of positive integers as sums of squares and polygonal numbers, linking these counts to Fourier coefficients of weight two quasimodular forms.

## Contribution

It introduces new formulas for weighted sums of representations of integers as sums of squares and polygonal numbers, connecting them to quasimodular forms.

## Key findings

- Formulas for weighted counts of sums of squares and polygonal numbers
- Connection between weighted sums and Fourier coefficients of quasimodular forms
- Extension to arbitrarily many summands

## Abstract

In this paper, we obtain formulas for the number of representations of positive integers as sums of arbitrarily many squares (and other polygonal numbers) with a certain natural weighting. The resulting weighted sums give Fourier coefficients of weight two quasimodular forms.