Positive definite quadratic forms representing integers of the form an 2+b

For any subset S of positive integers, a positive definite integral quadratic form is said to be S-universal if it represents every integer in the set S. In this article, we classify all binary S-universal positive definite integral quadratic forms in the case when S=S (a) ={an(2)vertical bar n >= 2} or S=S (a,b) ={an(2) + b vertical bar n is an element of Z}, where a is a positive integer and ab is a square-free positive integer in the latter case. We also prove that there are only finitely many S (a) -universal ternary quadratic forms not representing a. Finally, we show that there are exactly 15 ternary diagonal S (1)-universal quadratic forms not representing 1.