Sums of triangular numbers and sums of squares

For non-negative integers a,b, and n, let N(a,b;n) be the number of representations n as a sum squares with coefficients 1 or 3 (a ones b threes). Let N⁎(a,b;n) odd We have that N⁎(a,b;8n+a+3b) is triangular numbers It known for satisfying 1≤a+3b≤7, we haveN⁎(a,b;8n+a+3b)=22+(a4)+abN(a,b;8n+a+3b) a+3b=8, haveN⁎(a,b;8n+a+3b)=22+(a4)+ab(N(a,b;8n+a+3b)−N(a,b;(8n+a+3b)/4)). Such identities are not a+3b>8. In this paper, general a+b even, prove asymptotic equivalence formulas similar to above, n→∞. One our main results extends theorem Bateman, Datskovsky, Knopp where case b=0 was considered. Our approach different from Bateman-Datskovsky-Knopp's proof circle method singular series were used. achieve by explicitly computing Eisenstein components generating functions N(a,b;8n+a+3b). The use robust can adapted in studying asymptotics other representation coefficients.