A closed formula for the number of inequivalent ordered integer quadrilaterals with fixed perimeter

Document Type : Research Paper

Author

Faculty of Mathematics, University of Sciences and Technology Houari Boumediene, P.B. 32 El-Alia, 16111, Bab Ezzouar Algiers, Algeria

Abstract

Given an integer $n\geq4$, how many inequivalent quadrilaterals with ordered integer sides and perimeter $n$ are there? Denoting such number by $Q(n)$, the answer is given by the following closed formula:
\[
Q(n)=\left\{ \dfrac{1}{576}n\left( n+3\right) \left( 2n+3\right) -\dfrac{\left( -1\right) ^{n}}{192}n\left( n-5\right) \right\} \cdot
\]

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References

[1] G. E. Andrews, A note on partitions and triangles with integer sides, Amer. Math. Monthly, 86 (1979) 477–478.
[2] G. E. Andrews and K. Eriksson, Integer partitions, Cambridge University Press, Cambridge, 2004
[3] J. East and R. Niles, Integer polygons of given perimeter, Bull. Aust. Math. Soc., 100 (2019) 131–147.
[4] R. Honsberger, Mathematical gems. III, The Dolciani Mathematical Expositions, 9, Mathematical Association of America, Washington, DC, 1985.
[5] J. H. Jordan, R. Walch and R. J. Wisner, Triangles with integer sides, Amer. Math. Monthly, 86 (1979) 686–689.
[6] N. J. A. Sloane, The Online Encyclopedia of Integer Sequences, (2018), published electronically at http://oeis.org/.
Transactions on Combinatorics
Volume 13, Issue 4 - Serial Number 4
December 2024
Pages 327-334

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History

  • Receive Date: 21 February 2023
  • Revise Date: 01 August 2023
  • Accept Date: 03 August 2023
  • Published Online: 01 December 2024

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