Mixed sums of triangular numbers and certain binary quadratic forms

In this paper, we prove that for $d=3,\dots ,8$, every natural number can be written as $t_x+t_y+3t_z+d{t}_w$, where $x$, $y$, $z$, and $w$ are nonnegative integers and $t_k=k(k+1)\mathrm{/}2$ $(k=0,1,2,\dots )$ is a triangular number. Furthermore, we study mixed sums of triangular numbers and certain binary quadratic forms.