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The Frobenius Number for Sequences of Triangular Numbers Associated with Number of Solutions

Abstract

The famous linear diophantine problem of Frobenius is the problem to determine the largest integer (Frobenius number) whose number of representations in terms of \(a_1,\dots ,a_k\) is at most zero, that is not representable. In other words, all the integers greater than this number can be represented for at least one way. One of the natural generalizations of this problem is to find the largest integer (generalized Frobenius number) whose number of representations is at most a given nonnegative integer p. It is easy 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 three variables. In this paper we are successful to show explicit forms of the generalized Frobenius numbers of the triples of triangular numbers. When \(p=0\), their Frobenius number is given by Robles-Pérez and Rosales in 2018.

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Acknowledgements

The author thanks the anonymous reviewers for their careful reading of the manuscript and their insightful comments and suggestions.

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Correspondence to Takao Komatsu.

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The author declares to have no conflict of interests.

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Communicated by Frédérique Bassino.

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Komatsu, T. The Frobenius Number for Sequences of Triangular Numbers Associated with Number of Solutions. Ann. Comb. (2022). https://doi.org/10.1007/s00026-022-00594-3

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  • DOI: https://doi.org/10.1007/s00026-022-00594-3

Keywords

  • Frobenius problem
  • Frobenius numbers
  • Number of representations
  • Triangular numbers

MSC Classification

  • Primary 11D07
  • Secondary 05A15
  • 05A17
  • 05A19
  • 11B68
  • 11D04
  • 11P81