Mixed sums of triangular numbers and certain binary quadratic forms

TL;DR

This paper proves that for specific values of d, every natural number can be expressed as a sum involving triangular numbers and a scaled triangular number, and explores mixed sums with quadratic forms.

Contribution

It establishes universal representations for numbers using mixed sums of triangular numbers and quadratic forms for d=3 to 8, extending classical results.

Findings

Universal representation for d=3 to 8

Representation involving triangular numbers and quadratic forms

Extension of classical sum-of-figures results

Abstract

In this paper, we prove that for $d=3,\dots,8$, every natural number can be written as $t_x+t_y+3t_z+dt_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)/2$ $(k=0,1,2,\ldots)$ is a triangular number. Furthermore, we study mixed sums of triangular numbers and certain binary quadratic forms.