A universal sequence of integers generating balanced Steinhaus figures modulo an odd number |
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Authors: | Jonathan Chappelon |
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Affiliation: | a Univ. Lille Nord de France, F-59000 Lille, France b ULCO, LMPA J. Liouville, B.P. 699, F-62228 Calais, France c CNRS, FR 2956, France |
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Abstract: | In this paper, we partially solve an open problem, due to J.C. Molluzzo in 1976, on the existence of balanced Steinhaus triangles modulo a positive integer n, that are Steinhaus triangles containing all the elements of Z/nZ with the same multiplicity. For every odd number n, we build an orbit in Z/nZ, by the linear cellular automaton generating the Pascal triangle modulo n, which contains infinitely many balanced Steinhaus triangles. This orbit, in Z/nZ, is obtained from an integer sequence called the universal sequence. We show that there exist balanced Steinhaus triangles for at least 2/3 of the admissible sizes, in the case where n is an odd prime power. Other balanced Steinhaus figures, such as Steinhaus trapezoids, generalized Pascal triangles, Pascal trapezoids or lozenges, also appear in the orbit of the universal sequence modulo n odd. We prove the existence of balanced generalized Pascal triangles for at least 2/3 of the admissible sizes, in the case where n is an odd prime power, and the existence of balanced lozenges for all admissible sizes, in the case where n is a square-free odd number. |
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Keywords: | Molluzzo problem Balanced Steinhaus figure Universal sequence Steinhaus figure Steinhaus triangle Pascal triangle |
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