Consider the Fibonacci sequence ¹F n º 1 nD0 having initial conditions F 0 D 0, F 1 D 1 and recurrence relation F n D F n 1 CF n 2 .n 2/. The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recurrence relation. In this article, we study a new generalization ¹q n º, with initial conditions q 0 D 0 and q 1 D 1 which is generated by the recurrence relation q n D aq n 1 C q n 2 (when n is even) or q n D bq n 1 C q n 2 (when n is odd), where a and b are nonzero real numbers. Some well-known sequences are special cases of this generalization. The Fibonacci sequence is a special case of ¹q n º with a D b D 1. Pell's sequence is ¹q n º with a D b D 2 and the k-Fibonacci sequence is ¹q n º with a D b D k. We produce an extended Binet's formula for the sequence ¹q n º and, thereby, identities such as Cassini's, Catalan's, d'Ocagne's, etc.

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.

Given a set A consisting of positive integers a 1 < a 2 < • • • < a k and a k-term partition P : n 1 + n 2 + • • • + n k = n, find the extremal denominators of the regular and semi-regular continued fraction [0; x 1 , x 2 , . . . , x n ] with partial quotients x i ∈ A and where each a i occurs precisely n i times in the sequence x 1 , x 2 , . . . , x n . In 1983, G. Ramharter gave an explicit description of the extremal arrangements of the regular continued fraction and the minimizing arrangement for the semi-regular continued fraction and showed that in each case the arrangement is unique up to reversal and independent of the actual values of the positive integers a i . However, the determination of the maximizing arrangement for the semi-regular continuant turned out to be more difficult. He showed that if |A| = 2, then the maximizing arrangement is unique (up to reversal) and depends only on the partition P and not on the values of the a i . He further conjectured that this should be true for general A with |A| ≥ 2. In this paper, we confirm Ramharter's conjecture for sets A with |A| = 3 and give an algorithmic procedure for constructing the maximizing arrangement. We also show that Ramharter's conjecture fails in general for sets with |A| ≥ 4 in that the maximizing arrangement is neither unique nor independent of the values of the digits in A. The central idea, as discovered by Ramharter, is that the extremal arrangements satisfy a strong combinatorial condition. In the context of bi-infinite binary words, this condition coincides with the Markoff property, discovered by A.A. Markoff in 1879 in his study of minima of binary quadratic forms. We show that this same combinatorial condition, in the framework of infinite words over a k-letter alphabet, is the fundamental characterizing property which describes the orbit structure of codings of points under a symmetric k-interval exchange transformation.

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