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