The hyperbolic Pascal triangle HPT 4,q (q ≥ 5) is a new mathematical construction, which is a geometrical generalization of Pascal’s arithmetical triangle. In the present study we show that a natural pattern of rows of HPT 4,5 is almost the same as the sequence consisting of every second term of the well-known Fibonacci words. Further, we give a generalization of the Fibonacci words using the h...

Conventional Non-Linear Feedback Shift Registers (NLFSRs) use the Fibonacci configuration in which the value of the first bit is updated according to some non-linear feedback function of previous values of other bits, and each remaining bit repeats the value of its previous bit. We show how to transform the feedback function of a Fibonacci NLFSR into several smaller feedback functions of indivi...

An exact formula was recently obtained for the spectral norms of the Lucas and Fibonacci Hankel matrices [1], and also for the Lucas and Fibonacci Toeplitz matrices [4]. These results put finishing touches on the works initiated in [2] and [3]. On another front, bounds were found for the spectral norms of k-Fibonacci and k-Lucas Toeplitz matrices [7]. In this paper, we present the exact value f...

It is well-known that a continued fraction is periodic if and only if it is the representation of a quadratic irrational α. In this paper, we consider the family of sequences obtained from the recurrence relation generated by the numerators of the convergents of these numbers α. These sequences are generalizations of most of the Fibonacci-like sequences, such as the Fibonacci sequence itself, r...

Recently F. M. Dekking conjectured the form of subword complexity function for Fibonacci–Thue–Morse sequence. In this note we prove his conjecture by purely computational means, using free software Walnut.

In this paper, we …nd families of (0; 1; 1) tridiagonal matrices whose determinants and permanents equal to the negatively subscripted Fibonacci and Lucas numbers. Also we give complex factorizations of these numbers by the …rst and second kinds of Chebyshev polynomials. 1. Introduction The well-known Fibonacci sequence, fFng ; is de…ned by the recurrence relation, for n 2 Fn+1 = Fn + Fn 1 (1.1...

Among the most intensively studied integer sequences are the Fibonacci and Lucas sequences. Both are instances of second order recurrences [8], both satisfying sk−2+sk−1 = sk for all integers k, but where the fibonacci sequence (fi) begins with f0 = 0 and f1 = 1, the Lucas sequence (li) has l0 = 2 and l1 = 1. Several authors have recently been interested in the singular values of Toeplitz, circ...

We establish some identities relating two sequences that are, as explained, related to the Tribonacci sequence. One of these sequences bears the same resemblance to the Tribonacci sequence as the Lucas sequence does to the Fibonacci sequence. Defining a matrix that we call Tribomatrix, which extends the Fibonacci matrix, we see that the other sequence is related to the sum of the determinants o...

Many authors have considered the problem of extending the Fibonacci sequence to arbitrary real or complex subscripts (cf. [1], [6], and references therein). Since the positive integers form a discrete subset of R the existence of multitudes of continuous functions f : R→ R such that f(n) = Fn for positive integers n is immediate and the question then becomes one of determining the various prope...

Sloane’s influential On-Line Encyclopedia of Integer Sequences is an indispensable research tool in the service of the mathematical community. The sequence A001611 listing the “Fibonacci numbers + 1” contains a very large number of references and links. The sequence A000071 for the “Fibonacci numbers −1” contains an even larger number. Strangely, resentment seems to prevail between the two sequ...