Báo Cáo Báo cáo hóa học ON THE GROWTH RATE OF GENERALIZED FIBONACCI NUMBERS DONNIELL

ON THE GROWTH RATE OF GENERALIZED FIBONACCI NUMBERS DONNIELL E. FISHKIND Received 1 May 2004

Let α(t) be the limiting ratio of the generalized Fibonacci numbers produced by summing along lines of slope t through the natural arrayal of Pascal’s triangle. We prove that √ α(t) 3+t is an even function.

1. Overview Pascal’s triangle may be arranged in the Euclidean plane by associating the binomial coefficient ij with the point 1 3 j ư i, ư i ∈ R2 2 2 √ (1.1) for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure

1.1. The points in R2 associated with ij , i+1 , and i+1 form a unit equilateral triangle. This arrayal is j j+1 called the natural arrayal of Pascal’s triangle in R2 . √ √ For all t ∈ R : ư 3