Abstract: | Let $Bbb Z$ and $Bbb N$ be the set of integers and the set of positive
integers, respectively. For
$a,b,c,d,ninBbb N$ let $t(a,b,c,d;n)$ be the number of representations of
$n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2
+dw(w-1)/2$ $(x,y,z,winBbb Z$). In this paper we 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 $(1,3,3,4)$. |