Representations of integers by certain $2k$-ary quadratic forms

TL;DR

This paper derives formulas for counting how many integers can be expressed using specific $2k$-ary quadratic forms involving sums of squares with coefficients 1, 2, or 4.

Contribution

It provides explicit formulas for representations of integers by certain quadratic forms with coefficients 1, 2, or 4, extending understanding of these forms.

Findings

Formulas for representations with $l=2$ and $l=4$.

Explicit counts for integers represented by these forms.

Enhanced understanding of quadratic form representations.

Abstract

Suppose $k$ is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms $$ x_{1}^{2}+\cdots+x_{k}^{2}+l\left(x_{k+1}^{2}+\cdots+x_{2k}^{2}\right) $$ for $l\in\{2,4\}$.