Some Asymptotic Properties of Generalized Fibonacci Numbers

1. INTRODUCTION Horadam [1] has generalized two theorems of Subba Rao [3] which deal with some asymptotic p r o p e r t i e s of Fibonacci numbers. Horadam defined a sequence {w (n 2) where a , a are the roots of x 2-P 21 x + P 2 2 = 0. We shall let 06 """ LX r\ r\ UO r\-| • Horadam established two theorems for {w n }: I. The number of terms of {w n } not exceeding N is asymptotic to log(Nd/(P 22 w Q-a 2 1 w 1)). II. The range, within which the rank n of w n lies, is given by log w n + log (J-d) lx < n + 1 < log w n + log(J-d) /x, where X = y/(w_ 1 + 2x), J = y/(w_ 1-2x), X = W 0 " a 2 2 ^-l 5 2/ = W 0 " a 2 1 ^-l s and in which log stands for logarithm to the base a P 1 ; v = 2 in this case. These were generalizations of two theorems which Subba Rao had proved for ifJ • fn = w n {l, 1; 1,-1), the ordinary Fibonacci numbers. It is proposed here to explore generalizations of the Horadam-Subba Rao theorems to sequences, the elements of which satisfy linear recurrence relations of artitrary order. To this end, we define {w^} i J = I with suitable initial values w^\ n = 0, 1, ..., r-1, and where the P r j are arbitrary integers. Thus, {w^} represents Horadam f s generalized sequence of integers. We can suppose then that