Regular octahedra in {0, 1, ..., n} ³

• Autorzy: Ionascu E I
• Języki publikacji: EN

Abstrakty

EN

In this paper we describe a procedure for calculating the number of regular octahedra, RO(n), which have vertices with coordinates in the set {0,1,...,n}. As a result, we introduce a new sequence in The Online Encyclopedia of Integer Sequences (A178797) and list the first one hundred terms of it. We improve the method appeared in [12] which was used to find the number of regular tetrahedra with coordinates of their vertices in {0,1,..., n}. A new fact proved here helps increasing considerably the speed of all programs used before. The procedure is put together in a series of commands written for Maple and it is included in an earlier version of this paper in the matharxiv. Our technique allows us to find a series of cubic polynomials ...[wzór]

Słowa kluczowe

EN

diophantine equations; regular octahedra; Ehrhart polynomial; sequence of integers; primitive solutions

• Wydawca: Wydawnictwo Politechniki Poznańskiej
• Czasopismo: Fasciculi Mathematici
• Rocznik: 2012
• Tom: Nr 48
• Strony: 49--59
• Opis fizyczny: Bibliogr. 20 poz., fig.

Twórcy

autor

Ionascu E I

• Ionascu Department of Mathematics Columbus State University 4225 University Avenue Columbus, GA 31907

Bibliografia

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• [3] Chandler R., Ionascu E.J., A characterization of all equilateral triangles in Z³, Integers, Art. A19 of Vol. 8(2008).
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• [7].Grosswald E., Representations of Integers as Sums of Squares, Springer Verlag, New York, (1985).
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• [9] Hirschhorn M.D., Sellers J.A., On representations of numbers as a sum of three squares, Discrete Mathematics, 199(1999), 85-101.
• [10] Ionascu E.J., A parametrization of equilateral triangles having integer coordinates, Journal of Integer Sequences, 10(09.6.7.)(2007).
• [11] Ionascu E.J., Counting all equilateral triangles in {0,1, 2,..., n}³, Acta Math. Univ. Comenianae, LXXVII, 1(2008), 129-140.
• [12] Ionascu E.J., Regular tetrahedra with integer coordinates of their vertices, Acta Math. Univ. Comenianae, LXXX, 2(2011), 161-170.
• [13] Ionascu E.J., Obando r., Cubes in {0,1, 2, ...,n}², to appear in Integers (2012), arXiv:1003.4569.
• [14] Ionascu E.J., A characterization of regular tetrahedra in Z², J. Number Theory, 129(2009), 1066-1074.
• [15] Ionascu E.J., Markov A., Platonic solids in Z², J. Number Theory, 131 (2011), 138-145.
• [16] Ionascu E.J., Counting all regular octahedrons in {0,1,...,n}², arXiv: 1007.1655vl.
• [17] Larrosa I., Solution to Problem 8, http://faculty.missouristate.edu/l/lesreid /POW08-03.html.
• [18] Ionascu E.J., Platonic lattice polytopes and their Ehrhart polynomial, (to appear).
• [19] Rosen K., Elementary Number Theory, Fifth Edition, Addison Wesley, 2004.
• [20] Sloane N.J.A., The On-Line Encyclopedia of Integer Sequences, 2005, published electronically at http://oeis.org/.