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The famous linear diophantine problem of Frobenius is the problem todetermine the largest integer (Frobenius number) whose number ofrepresentations in terms of $a_1,\dots,a_k$ is at most zero, that is notrepresentable. In other words, all the integers greater than this number can berepresented for at least one way. One of the natural generalizations of thisproblem is to find the largest integer (generalized Frobenius number) whosenumber of representations is at most a given nonnegative integer $p$. It iseasy to find the explicit form of this number in the case of two variables.However, no explicit form has been known even in any special case of threevariables. In this paper we are successful to show explicit forms of thegeneralized Frobenius numbers of the triples of triangular numbers. When $p=0$,their Frobenius number is given by Robles-P\'erez and Rosales in 2018.

The Frobenius number for sequences of triangular numbers associated with number of solutions