On Divisibility of Fibonomial Coefficients by 3
نویسنده
چکیده
Let Fn be the nth Fibonacci number. For 1 ≤ k ≤ m − 1 let
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The p-adic Order of Some Fibonomial Coefficients
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Let (Fn)n≥0 be the Fibonacci sequence. For 1 ≤ k ≤ m, the Fibonomial coefficient is defined as [ m k ] F = Fm−k+1 · · ·Fm−1Fm F1 · · ·Fk , and [ m k ] F = 0, for k > m. In this paper, we shall prove that if p is a prime number such that p ≡ −2 or 2 (mod 5), then p | [ p pa ]
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I. Up to our knowledge-since about 126 years we were lacking of classical type combinatorial interpretation of Fibonomial coefficients as it was Lukas [1]-to our knowledge-who was the first who had defined Finono-mial coefficients and derived a recurrence for them (see Historical Note in [2]). Namely as accurately noticed by Knuth and Wilf in [3] the recurrent relations for Fibonomial coefficie...
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Combinatorial interpretation of the fibonomial coefficients recently proposed by the present author [1,2] here results in combinatorial derivation of the recurrence relation for fibonomial coefficients . The presentation is provided with quite an exhaustive context with reference to classical attitude of [3,4]. This note apart from plane grid coordinate system used is fitted with several figure...
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It is well known that Pascal’s triangle exhibits fractal behavior when reduced modulo a prime. We show that the triangle of Fibonomial coefficients has a similar nature modulo two. Specifically, for any m ≥ 0, the subtriangle consisting of the first 3 · 2m rows is duplicated on the left and right sides of the next 3 · 2m rows, with an inverted triangle of zeros in between. We give three proofs ...
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