Universal mixed sums of generalized $4$- and $8$-gonal numbers

An integer of the form $P_m(x)= \frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers $\alpha_1,\dots,\alpha_u$ and $\beta_1,\dots,\beta_v$, a mixed sum $\Phi=\alpha_1P_4(x_1)+\cdots+\alpha_uP_4(x_u)+\beta_1P_8(y_1)+\cdots+\beta_vP_8(y_v)$ of generalized $4$- and $8$-gonal numbers is called universal if $\Phi=N$ has an integer solution for any nonnegative integer $N$. In this article, we prove that there are exactly 1271 proper universal mixed sums of generalized $4$- and $8$-gonal numbers. Furthermore, the "$61$-theorem" is proved, which states that an arbitrary mixed sum of generalized $4$- and $8$-gonal numbers is universal if and only if it represents the integers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$, $13$, $14$, $15$, $18$, $20$, $30$, $60$, and $61$.