On the number of representations of n by ax(x−1)/2+by(y−1)/2

Let N be the set of positive integers. For b,n∈N let tn(1,b) denote the number of representations 〈x,y〉 (x,y∈N) of n=x(x−1)/2+by(y−1)/2. In the paper we mainly obtain explicit formulas for tn(1,b) in the cases b=2,4,5,9,11,13,19,23,25,27,31,37,43,67,163.