## 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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## References

R. Apéry,

*Sur les branches superlinéaires des courbes algébriques*, C. R. Acad. Sci. Paris**222**(1946), 1198–1200.L. Bardomero and M. Beck,

*Frobenius coin-exchange generating functions*, Amer. Math. Monthly**127**(2020), no. 4, 308–315.M. Beck, I.M. Gessel, T. Komatsu, The polynomial part of a restricted partition function related to the Frobenius problem. Electron. J. Combin.

**8**(1), 7 (2001).M. Beck and C. Kifer,

*An extreme family of generalized Frobenius numbers*, Integers**11**(2011), A24, 639–645.A. Brauer and B. M. Shockley,

*On a problem of Frobenius*, J. Reine Angew. Math.**211**(1962), 215–220.T. C. Brown and P. J. Shiue,

*A remark related to the Frobenius problem*, Fibonacci Quart.**31**(1993), 32–36.A. Cayley, On a problem of double partitions. Philos. Mag.

**XX**(1860), 337–341.F. Curtis,

*On formulas for the Frobenius number of a numerical semigroup*, Math. Scand.**67**(1990), 190–192.T. Komatsu,

*On the number of solutions of the Diophantine equation of Frobenius–General case*, Math. Commun.**8**(2003), 195–206.T. Komatsu, The Frobenius number associated with the number of representations for sequences of repunits. C. R. Math. Acad. Sci. Paris (

**In press**)T. Komatsu,

*On*\(p\)-*Frobenius and related numbers due to*\(p\)-*Apéry set*, arXiv:2111.11021.T. Komatsu and Y. Zhang,

*Weighted Sylvester sums on the Frobenius set*, Irish Math. Soc. Bull.**87**(2021), 21–29.T. Komatsu and Y. Zhang,

*Weighted Sylvester sums on the Frobenius set in more variables*, Kyushu J. Math.**76**(2022), 163–175.A. M. Robles-Pérez and J. C. Rosales,

*The Frobenius number for sequences of triangular and tetrahedral numbers*, J. Number Theory**186**(2018), 473–492.E. S. Selmer,

*On the linear diophantine problem of Frobenius*, J. Reine Angew. Math.**293/294**(1977), 1–17.N. J. A. Sloane,

*The On-Line Encyclopedia of Integer Sequences*, available at oeis.org. (2022).J. J. Sylvester,

*On the partition of numbers*, Quart. J. Pure Appl. Math.**1**(1857), 141–152.J. J. Sylvester,

*On subinvariants, i.e. semi-invariants to binary quantics of an unlimited order*, Am. J. Math.**5**(1882), 119–136.J. J. Sylvester,

*Mathematical questions with their solutions*, Educational Times**41**(1884), 21.J. J. Tattersall,

*Elementary number theory in nine chapters*, Second ed. Cambridge Univ. Press, Cambridge, 2005. xii+430 pp. ISBN: 978-0-521-61524-2; 0-521-61524-0A. Tripathi,

*The number of solutions to*\(a x+b y=n\), Fibonacci Quart.**38**(2000), 290–293.A. Tripathi,

*On sums of positive integers that are not of the form*\(a x+b y\), Amer. Math. Monthly**115**(2008), 363–364.

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

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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