Abstract
Take any entry X inside Pascal’s triangle and divide the six entries surrounding X into two sets S 1 and S 2 by taking these six entries alternately. Then the product of three elements in S 1 is equal to one in S 2. This property was found by Hoggatt and Hansell [7], and called the hexagon property w.r.t. the product. The hexagon property w.r.t. GCD was also established by Hillman and Hoggatt [6], while this property does not hold w.r.t. LCM. S. Ando [1] proposed a modified number array that has (n + 1)!/h!(n — h)! as its entry, where the hexagon properties w.r.t. the product and LCM are valid but one w.r.t. GCD does not hold. In order to establish the basic principle for such properties, Ando and Sato [5] studied the center covering stars in Pascal’s triangle, the definition of which will be shown in the next section. Applying part of our results, we constructed in [3] translatable and rotatable configurations which give equal product, equal GCD and equal LCM properties simultaneously.
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References
Ando, S. “A Triangular Array with Hexagon property, Dual to Pascal’s Triangle.” Application of Fibonacci Numbers, Volume 2. Edited by Gerald E. Bergum, A.N. Philippou, and A.F. Horadam. Kluwer Academic Publishers, Dordrecht, The Netherlands, 1988: pp. 61–67.
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Ando, S. and Sato, D. “A Necessary and Sufficient Condition that Rays of a Star Configuration on Pascal’s Triangle Cover Its Center with Respect to GCD and LCM.” To appear.
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© 1993 Springer Science+Business Media Dordrecht
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Ando, S., Sato, D. (1993). On the Minimal Center Covering Stars with Respect to GCD in Pascal’s Pyramid and its Generalizations. In: Bergum, G.E., Philippou, A.N., Horadam, A.F. (eds) Applications of Fibonacci Numbers. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2058-6_3
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DOI: https://doi.org/10.1007/978-94-011-2058-6_3
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