  In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that
under the generalized Riemann hypothesis any positive odd integer greater than
2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$,
equivalently the form $2x^2+5y^2+4T_z$ represents all integers greater than
1359, where $T_z$ denotes the triangular number $z(z+1)/2$. Given positive
integers $a,b,c$ we employ modular forms and the theory of quadratic forms to
determine completely when the general form $ax^2+by^2+cT_z$ represents
sufficiently large integers and establish similar results for the forms
$ax^2+bT_y+cT_z$ and $aT_x+bT_y+cT_z$. Here are some consequences of our main
theorems: (i) All sufficiently large odd numbers have the form $2ax^2+y^2+z^2$
if and only if all prime divisors of $a$ are congruent to 1 modulo 4. (ii) The
form $ax^2+y^2+T_z$ is almost universal (i.e., it represents sufficiently large
integers) if and only if each odd prime divisor of $a$ is congruent to 1 or 3
modulo 8. (iii) $ax^2+T_y+T_z$ is almost universal if and only if all odd prime
divisors of $a$ are congruent to 1 modulo 4. (iv) When $v_2(a)\not=3$, the form
$aT_x+T_y+T_z$ is almost universal if and only if all odd prime divisors of $a$
are congruent to 1 modulo 4 and $v_2(a)\not=5,7,...$, where $v_2(a)$ is the
2-adic order of $a$.
