For non-negative integers $a$, $b$, and $n$, let $t(a,b;n)$ be the number of representations of $n$ as a sum of triangular numbers with coefficients $1$ and $3$ ($a$ of ones and $b$ of threes) and let $r(a, b; n)$ be the number of representations of $n$ as a sum of squares with coefficients $1$ and $3$ ($a$ of ones and $b$ of threes). It is known that for $a$ and $b$ satisfying $1\leq a+3b \leq 7$, we have $$ t(a,b;{n}) = \frac{2}{2+{a\choose4}+ab} r(a,b;8n+a+3b) $$ and for $a$ and $b$ satisfying $a+3b=8$, we have $$ t(a,b;{n}) = \frac{2}{2+{a\choose4}+ab} \left( r(a,b;8n+a+3b) - r(a,b; (8n+a+3b)/4) \right). $$ Such identities are not known for $a+3b>8$.

Such identities are not known for $a+3b>8$. We report on our joint work with Zafer Selcuk Aygin (University of Calgary) in which, 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 $t(a,b;n)$ and $r(a,b;n)$.