  For an integer $x$ let $t_x$ denote the triangular number $x(x+1)/2$.
Following a recent work of Z. W. Sun, we show that every natural number can be
written in any of the following forms with $x,y,z\in\Z$: $$x^2+3y^2+t_z,
x^2+3t_y+t_z, x^2+6t_y+t_z, 3x^2+2t_y+t_z, 4x^2+2t_y+t_z.$$ This confirms a
conjecture of Sun.
