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 $ax(x-1)/2+by(y-1)/2+cz(z-1)/2
+dt(t-1)/2$ $(x,y,z,tinBbb Z$). In this paper we obtain explicit formulas
for $t(a,b,c,d;n)$ in the cases
$(a,b,c,d)=(1,1,4,4), (1,4,4,4), (1,2,2,4), (1,2,4,4)$, $(1,1,9,9),
(1,9,9,9), (1,1,1,9), (1,3,9,9), (1,1,3,9).$ |