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 of n as a sum of squares with coefficients 1 or 3 (a of ones and b of threes). Let N*(a, b; n) be the number of representations of n as a sum of odd squares with coefficients 1 or 3 (a of ones and b of threes). We have that N*(a, b; 8n + a + 3b) is the number of representations of n as a sum of triangular numbers with coefficients 1 or 3 (a of ones and b of threes). It is known that for a and b satisfying 1 <= a + 3b <= 7, we have N*(a, b; 8n + a + 3b) = 2/2 + (a4) +ab N(a, b; 8n + a + 3b) and for a and b satisfying a + 3b = 8, we have N*(a, b; 8n + a + 3b) = 2/2+(a4) +ab (N(a, b; 8n + a + 3b) - N(a, b; (8n + a + 3b)/4)) . Such identities are not known for a+3b > 8. In this paper, for general a and b with a +b even, we prove asymptotic equivalence of formulas similar to the above, as n-+ infinity. One of our main results extends a theorem of Bateman, Datskovsky, and Knopp where the case b = 0 and general a was considered. Our approach is different from Bateman-Datskovsky-Knopp's proof where the circle method and singular series were used. We achieve our results by explicitly computing the Eisenstein components of the generating functions of N*(a, b; 8n + a + 3b) and N(a, b; 8n + a + 3b). The method we use is robust and can be adapted in studying the asymptotics of other representation numbers with general coefficients. (c) 2022 Elsevier Inc. All rights reserved.