On the number of representations of n as a linear combination of four triangular numbers II

Original · EN

Let Z and N be the set of integers and the set of positive integers, respectively. For a,b,c,d,n∈ N let N(a,b,c,d;n) be the number of representations of n by ax²+by²+cz²+dw², and let t(a,b,c,d;n) be the number of representations of n by ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2 (x,y,z,w∈ Z). In this paper we reveal the connections between t(a,b,c,d;n) and N(a,b,c,d;n). Suppose a,n∈ N and 2 a. We show that t(a,b,c,d;n)= 23N(a,b,c,d;8n+a+b+c+d)-2N(a,b,c,d;2n+(a+b+c+d)/4) for (a,b,c,d)= (a,a,2a,8m),(a,3a,8k+2,8m+6),(a,3a,8m+4,8m+4)(n≡ m+a-12 2) and (a,3a,16k+4,16m+4)(n≡ a-12 2). We also obtain explicit formulas for t(a,b,c,d;n) in the cases (a,b,c,d)=(1,1,2,8),(1,1,2,16),(1,2,3,6),(1,3,4,12),(1,1, 3,4),(1,1,5,5),(1,5,5,5),(1,3,3,12),(1,1,1,12),(1,1,3,12) and (1,3,3,4).