ON THE GROWTH RATE OF GENERALIZED FIBONACCI NUMBERS

for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure 1.1. The points in R2 associated with ( i j ) , ( i+1 j ) , and ( i+1 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 k(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 ( k 0 ) . It is well known that { k( √ 3/3)}k=0 is the Fibonacci sequence F0,F1,F2, . . . , and { k(− √ 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 { k(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


Overview
Pascal's triangle may be arranged in the Euclidean plane by associating the binomial coefficient i j with the point for all nonnegative integers i, j such that j ≤ i, as illustrated in Figure 1.1. The points in R 2 associated with i j , i+1 j , and i+1 j+1 form a unit equilateral triangle. This arrayal is called the natural arrayal of Pascal's triangle in R 2 .
For all t ∈ R : − √ 3 < t < √ 3 and nonnegative integers k, define ᏸ k (t) to be the sum of all binomial coefficients associated with points in R 2 which are on the line of slope t through the point in R 2 associated with k 0 . It is well known that {ᏸ k ( √ 3/3)} ∞ k=0 is the Fibonacci sequence F 0 ,F 1 ,F 2 ,..., and {ᏸ k (− √ 3/3)} ∞ k=0 is the sequence of every other Fibonacci number F 0 ,F 2 ,F 4 ,..., as illustrated in Figure 1.1; for a fixed t, the sequence {ᏸ k (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 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) := lim k→∞ ᏸ k+1 (t)/ᏸ k (t). The following is our main result.
(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 in Section 2. In the setting of the left-justified arrayal, Harris and Styles (and, effectively, Dickinson) show that generalized Fibonacci numbers satisfy the difference equation (2.5) in Section 2, and thus have rate of growth as given in (2.6) of Section 2. Ferguson [3] investigated the roots of the polynomial in (2.6) when q is an integer.
Our contribution is to investigate this rate of growth as a function of the generating slope, to transfer the setting to the natural arrayal of Pascal's triangle, and, in Section 3, to prove Theorem 1.1. In Section 2, we review classical facts and correlate them to the natural arrayal of Pascal's triangle.

The Left-Justified Arrayal
It is sometimes easier to consider the left-justified arrayal of Pascal's triangle in R 2 , in which the binomial coefficient i j is associated with the point ( j,−i) ∈ R 2 for all nonnegative integers i, j : j ≤ i, as illustrated in Figure 2 Lines of slope q = n/d in our left-justified arrayal correspond to lines of slope in the natural arrayal, and k 0 is a summand of L kd (q) and thus, for all nonnegative integers k, L kd (q) = ᏸ k ( √ 3q/(q + 2)). Hence, 3), we write q as a function of t, obtaining, by (2.3), For the moment, suppose that q is positive. Using the identity i j+1 , there is a correspondence (see Endnote(I)) between the binomial coefficients summed in L k (q), those summed in L n+k (q), and those summed in L n+d+k (q) yielding the linear difference equation It is not hard to verify that the associated auxiliary polynomial x n+d − x n − 1 has distinct roots, say λ 1 ,λ 2 ,...,λ n+d , and the initial conditions ensure nonzero constants c 1 ,c 2 ,..., c n+d ∈ R in the expansion L k (q) = n+d l=1 c l λ k l , k = 0,1,2,... (these constants c l are given explicitly in [2]). Among these roots, λ 1 ,λ 2 ,...,λ n+d , there is a unique positive root, and this root is also the root of maximum modulus (see Endnote(II)). Thus, β(q) is the unique positive root of x n+d − x n − 1 and, substituting (2.1) into this, γ(q) is the unique positive root of x q+1 − x q − 1.
Proof. Suppose q = n/d > −1 such that n and d are relatively prime integers and d is positive. Note that −n/(n + d) > −1 and observe that β(n/d) and β(−n/(n + d)) are each the unique positive root of x n+d − x n − 1, which is also the unique positive root of x d − x −n − 1 and so, in particular, Taking the dth root of (3.2) and simplifying yields the desired result.
We next prove our main result, Theorem 1.1, which states that for all t ∈ R such that − √

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall. This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/ according to the following timetable: Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009