Formulas for the Generalized Frobenius Number of Triangular Numbers

Abstract

For $ k \geq 2 $, let $ A = (a_{1}, a_{2}, \ldots, a_{k}) $ be a $k$-tuple of positive integers with $\gcd(a_{1}, a_2, \ldots, a_k) = 1$. For a non-negative integer $s$, the generalized Frobenius number of $A$, denoted as $\mathtt{g}(A;s) = \mathtt{g}(a_1, a_2, \ldots, a_k;s)$, represents the largest integer that has at most $s$ representations in terms of $a_1, a_2, \ldots, a_k$ 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 $s \geq 0$ where $t_n$ is given by $\binom{n+1}{2}$. Furthermore, we present the proof of Komatsu's conjecture