A GCD Property on Pascal’s Pyramid and the Corresponding LCM Property of the Modified Pascal Pyramid

Shiro Ando and Daihachiro Sato

A chapter in Applications of Fibonacci Numbers, 1990, pp 7-14 from Springer

Abstract: Abstract Concerning the six binomial coefficients A1, A2,…, A6 A2 A3 surrounding any entry A inside Pascal’s triangle, Hoggatt and A1 A A4 Hansell [1] proved the identity (1) $${A_1}{A_3}{A_5} = {A_2}{A_4}{A_6},$$ which has been generalized to the case of multinomial coefficients by Hoggatt and Alexanderson [2]. Meanwhile, Gould [3] found the remarkable property (2) $$\gcd \left( {{A_1},{A_3},{A_5}} \right) = \gcd \left( {{A_2},{A_4},{A_6}} \right),$$ which was established by Hillman and Hoggatt [4] for the generalized binomial coefficients defined by (16) for m=2. He also showed that the equality (3) $$lcm\left( {{A_1},{A_3},{A_5}} \right) = lcm\left( {{A_2},{A_4},{A_6}} \right)$$ does not always hold.

Keywords: Multinomial Coefficient; Binomial Coefficient; Fibonacci Sequence; Triangular Array; Integral Coefficient (search for similar items in EconPapers)
Date: 1990
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DOI: 10.1007/978-94-009-1910-5_2

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