On the number of representations of n as a linear combination of four triangular numbers II

TL;DR

This paper explores the relationship between the number of representations of an integer as a sum of four triangular numbers and as a quadratic form, providing explicit formulas and connections for specific parameter cases.

Contribution

It establishes formulas linking $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$, revealing new connections and explicit representations for particular parameter sets.

Findings

Derived formulas connecting $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$

Explicit formulas for specific parameter cases

Identified congruence conditions for representations

Abstract

Let $\Bbb Z$ and $\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in\Bbb Z$). In this paper we reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in\Bbb N$ and $2\nmid a$. We show that $$t(a,b,c,d;n)=\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$ for $(a,b,c,d)= (a,a,2a,8m),\ (a,3a,8k+2,8m+6),\ (a,3a,8m+4,8m+4)\ (n\equiv m+\frac{a-1}2 \pmod 2)$ and $(a,3a,16k+4,16m+4)\ (n\equiv \frac{a-1}2\pmod 2)$. We also obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,1,2,8),\ (1,1,2,16),(1,2,3,6),\ (1,3,4,12),\ (1,1,$ $3,4),\ (1,1,5,5),\ (1,5,5,5),\ (1,3,3,12),\ (1,1,1,12),\ (1,1,3,12)$ and $(1,3,3,4)$.