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

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 $t(a,b,c,d;n)$ 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 obtain explicit formulas for $t(a,b,c,d;n)$ in the cases $(a,b,c,d)=(1,2,2,4),\ (1,2,4,4),\ (1,1,4,4),\ (1,4,4,4)$, $(1,3,9,9),\ (1,1,3,9)$, $(1,3,3,9)$, $(1,1,9,9),\ (1,9,9,9)$ and $(1,1,1,9).$