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

Let $\mathbb{Z}$ and $\mathbb{N}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{N}$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $a{x}^2+b{y}^2+c{z}^2+d{w}^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\in \mathbb{Z}$). In this paper we reveal the connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\in \mathbb{N}$ and $2\nmid a$\mathrm{.}Weshowthat$$t(a,b,c,d;n)=\frac 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4)$$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$(1,3,3,4)$.