Formulas for the Generalized Frobenius Number of Triangular Numbers

For $k\ge 2$, let $A=(a_1,a_2,\dots ,a_k)$ be a $k$-tuple of positive integers with $\gcd (a_1,a_2,\dots ,a_k)=1$. For a non-negative integer $s$, the generalized Frobenius number of $A$, denoted as $\mathtt{g}(A;s)=\mathtt{g}(a_1,a_2,\dots ,a_k;s)$, represents the largest integer that has at most $s$ representations in terms of $a_1,a_2,\dots ,a_k$ with non-negative integer coefficients. In this article, we provide a formula for the generalized Frobenius number of three consecutive triangular numbers, $\mathtt{g}(t_n,t_{n+1},t_{n+2};s)$, valid for all $s\ge 0$ where $t_n$ is given by $\left(\genfrac{}{}{0px}{}{n+1}{2}\right)$. Furthermore, we present the proof of Komatsu’s conjecture