The <i>p</i> -Frobenius number for the triple of certain quadratic numbers

Abstract In this paper, we give closed-form expressions of the $p$-Frobenius number for the triple of the numbers $a n(n-1)+r$ for an integer $a\ge 4$ and $r$ is odd. For the set of given positive integers $A:=\{a_{1},a_{2},\dots ,a_{k}\}$, the $p$-Frobenius number is the largest integer whose nonnegative integral linear combinations of given positive integers in $A$ are expressed in at most $p$ ways. When $p=0$, the $0$-Frobenius number is the classical Frobenius number, which is the central topic of the famous linear Diophantine problem of Frobenius.