Some new relations between T(a1,a2,a3,a4,a5;n) and N(a1,a2,a3,a4,a5;n)

Let $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$ count the representations of $n$ as $a_1x_1^2+a_2x_2^2+a_3x_3^2+a_4x_4^2+a_5x_5^2$ and $a_1X_1(X_1+1)/2+a_2X_2(X_2+1)/2+a_3X_3(X_3+1)/2+a_4X_4(X_4+1)/2+a_5X_5(X_5+1)/2$, respectively, where $a_1,a_2,a_3,a_4,a_5$ are positive integers, $x_1,x_2,x_3,x_4,x_5$ are integers and $n,X_1,X_2,X_3,X_4,X_5$ are nonnegative integers. In this paper, we establish some new relations between $N(a_1,a_2,a_3,a_4,a_5;n)$ and $T(a_1,a_2,a_3,a_4,a_5;n)$. Also, we prove that $T(a_1,a_2,a_3,a_4,a_5;n)$ is a linear combination of $N(a_1,a_2,a_3,a_4,a_5;m)$ and $N(a_1,a_2,a_3,a_4,a_5;m/4)$, where $m=8n+a_1+a_2+a_3+a_4+a_5$, for various values of $a_1,a_2,a_3,$ $a_4,a_5$.