The k-Periodic Fibonacci Sequence and an Extended Binet's Formula

Abstract

It is well known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational˛. In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers˛. These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r-Fibonacci sequences, and the Pell sequence, to name a few. We show that these sequences satisfy a linear recurrence relation when considered modulo k, even though the sequences themselves do not. We then employ this recurrence relation to determine the generating functions and Binet-like formulas. We end by discussing the convergence of the ratios of the terms of these sequences.

A NOTE ON THE MODIFIED k-FIBONACCI-LIKE SEQUENCE

“…Binet's formulae are well known in the study of sequences like Fibonacci sequence [1,2,3,4,6,7,8,10,11,12]. In this section, we introduce and prove Binet's formula for the modified k-Fibonacci-like sequence.…”

“…Edson and Yayenie [4] introduced the generalized Fibonacci sequence and proved some related identities. For any two nonzero real numbers a and b, the generalized Fibonacci sequence {q n } ∞ n=0 is defined by the recurrence relation (3) q n = aq n−1 + q n−2 , if n is even, bq n−1 + q n−2 , if n is odd, (n ≥ 2), with q 0 = 0 and q 1 = 1.…”

“…Edson, Lewis and Yayenie [3] introduced the k-periodic Fibonacci sequence and proved some related identities. For any k-tuple (x 1 , x 2 , .…”

SOME PROPERTIES OF THE GENERALIZED FIBONACCI SEQUENCE {qn} BY MATRIX METHODS

“…These sequences arise in a natural way in the study of continued fractions of quadratic irrationals (see [3]) and combinatorics on words or dynamical system theory. Some well-known sequences are special cases of this generalization.…”

General conditional recurrences

“…We call such a sequence a general conditional recurrence sequence with associated coefficient matrix (2) If s ¼ 2 and a i;2 ¼ 1 for 0 6 i 6 r À 1 with any nonzero real numbers a i;1 , we obtain the sequences studied in [3], the k-periodic Fibonacci sequences. These sequences are also studied independently in [11,13].…”