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 $1\leq a+3b \leq 7$, we have $$N^*(a,b;8n+a+3b)= \frac{2}{2+{a\choose4}+ab} N(a,b;8n+a+3b)$$ and for $a$ and $b$ satisfying $a+3b=8$, we have $$N^*(a,b;8n+a+3b) = \frac{2}{2+{a\choose4}+ab} \left( N(a,b;8n+a+3b) - N(a,b; (8n+a+3b)/4) \right).$$ 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 $n\rightarrow\infty$. 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.