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 − i, − i ∈ R2 2 2 √ () for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure . The points in R2 associated. | ON THE GROWTH RATE OF GENERALIZED FIBONACCI NUMBERS DONNIELL E. FISHKIND Received 1 May 2004 Let a 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 a t 3 t is an even function. 1. Overview Pascal s triangle may be arranged in the Euclidean plane by associating the binomial coefficient j with the point U_1 y3 G R2 V - 2i - 2 7 e R for all nonnegative integers i j such that j i as illustrated in Figure . The points in R2 associated with i 1 and i 1 form a unit equilateral triangle. This arrayal is called the natural arrayal of Pascal s triangle in R2. For all t e R - 3 t 73 and nonnegative integers k define ẩk t to be the sum of all binomial coefficients associated with points in R2 which are on the line of slope t through the point in R2 associated with 0 . It is well known that k 73 3 k 0 is the Fibonacci sequence F0 F1 F2 . and k -73 3 k Q is the sequence of every other Fibonacci number F0 F2 F4 . as illustrated in Figure for a fixed t the sequence k t k Q is called the generalized Fibonacci sequence induced by the slope t. Generalized Fibonacci numbers arise in many ways for example for any integers a b 1 b a the number of ways to distribute a identical objects to any number of distinct recipients such that each recipient receives at least b objects is I l -1 1 1 a - l b b - 1 l - 1 a-b b v5 . Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 4 2004 273-277 2000 Mathematics Subject Classification 39A11 11B39 11B65 URL http S1687183904310034 274 Growth rate of generalized Fibonacci numbers i ị ị ị 1 2 5 13 Figure . The natural arrayal of Pascal s triangle and Fibonacci numbers as line sums. For all t e R - 3 t V3 we define a t to be the limiting ratio of the generalized Fibonacci sequence induced by the slope t that is a t limk -vẩk 1 t ẩk t . The following is our main result.
