Sums of triangular numbers

The triangular numbers are $t_k=k(k+1)/2,$ $(k=0,1,2,...).$ In this paper, we prove that for $d=3,4,5,6,7,8,$ every natural number can be written as $t_x+t_y+3t_z+dt_w,$ where $x,y,z,w$ are all nonnegative integers. Furthermore, we show the applications of this fact.