On the growth rate of generalized Fibonacci numbers
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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 co efficient ij with the point
√
1 3 j − i, − 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 i+1 R2 associated with ij , i+1 , and j j+1 form a unit equilateral triangle. This arrayal is called the natural arrayal of Pascal’s triangle in R2 . √ √ For all t ∈ R : − 3 < t < 3 and nonnegative integers k, define ᏸk (t) to be the sum of all binomial coefficients associated with points in R2 which are on the line of slope √ k 2 t through the point in R associated with 0 . It is well known that {ᏸk ( 3/3)}∞ k=0 is √ ∞ the Fibonacci sequence F0 ,F1 ,F2 ,..., and {ᏸk (− 3/3)}k=0 is the sequence of every other Fibonacci number F0 ,F2 ,F4 ,..., as illustrated in Figure 1.1; for a fixed t, the sequence {ᏸk (t)}∞ k=0 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 ∞ l−1+a−l·b b −1√ 3 . = ᏸa−b l−1 b+1 l=1 Copyright © 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004:4 (2004) 273–277 2000 Mathematics Subject Classification: 39A11, 11B39, 11B65 URL: http://dx.doi.org/10.1155/S1687183904310034
(1.2)
274
Growth rate of generalized Fibonacci numbers y 0
x
1
1
0
1
2 2 2 0
1
2
3 3 3 3 0
1
3
2
4 4 4 4 4 0
1
3
2
√
4
0
1
2
3
4
1 1 0
1
3 3 0
2
3 3
1
0
1 1
3
2
4 4 4 0
5
1
2 2 2 0
4
4
3
4
2
√
3 =1 3 √ 3 ᏸ1 − =2 3 √ 3 ᏸ2 − =5 3 √ 3 ᏸ3 − = 13 3 ᏸ0 −
0
5 5
5 5 5 5 5 5
0
3 =2 3 √ 3 ᏸ3 =3 3 √ 3 ᏸ4 =5 3 √ 3 ᏸ5 =8 3 ᏸ2
0
5 5 5 5 2
3
4
5
6 6
Figure 1.1. The natural arrayal of Pascal’s triangle and Fibonacci numbers as line sums.
√
√
For all t ∈ R : − 3 < t < 3, we define α(t) to be the limiting ratio of the generalized Fibonacci sequence induced by the slope t; that is, α(t) := limk→∞ ᏸk+1 (t)/ᏸk (t). The following is our main result. √
√
Theorem 1.1. For all t ∈ R : − 3 < t < 3, it holds that α(t) √
√
3+t
= α(−t)
√
3−t .
(Theorem 1.1 is easily and directly verified when t = ± 3/3, since the rate of growth of the sequence of every other Fibonacci number is the square of the rate of growth of the Fibonacci sequence.) Generalized Fibonacci numbers arising as line sums through Pascal’s triangle were introduced by Dickinson [2], Harris and Styles [4], and Hochster [6], and have been presented extensively in the literature (see [1, 5, 7]). The classical setting has been the left-justified arrayal of Pascal’s triangle, which we define
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