相关论文: G\'en\'eralisation du Th\'eor\`eme de Zeckendorf

In this paper we find a Clifford algebra associated to generalized Fibonacci quaternions. In this way, we provide a nice algorithm to obtain a division quaternion algebra starting from a quaternion non-division algebra and vice-versa.

The object of this paper is to generalize a theorem on the binomial coefficient [4] to the case in an arithmetic progression. We will also give a slightly stronger result than Langevin's [2].

In this note, we obtain some identities for the generalized Fibonacci polynomial by using the Q(x) matrix. These identities including the Cassini identity and Honsberger formula can be applied to some polynomial sequences, such as Fibonacci…

For $k\geq 2$, the $k$-generalized Fibonacci sequence $(F_n^{(k)})_{n}$ is defined by the initial values $0,0,\ldots,0,1$ ($k$ terms) and such that each term afterwards is the sum of the $k$ preceding terms. In this paper, we search for…

In this article, we present a trick around Fibonacci numbers which can be found in several magic books. It consists in computing quickly the sum of the successive terms of a Fibonacci-like sequence. We give explanations and extensions of…

In this paper, algorithms are developed for computing the Stirling transform and the inverse Stirling transform; specifically, we investigate a class of sequences satisfying a two-term recurrence. We derive a general identity which…

We present numerous interesting, mostly new, results involving the $n$-step Fibonacci numbers and $n$-step Lucas numbers and a generalization. Properties considered include recurrence relations, summation identities, including binomial and…

Zeckendorf's theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers; this result has been generalized to many recurrence relations, especially those arising from linear recurrences with…

Trace maps of two-letter substitution rules are investigated with special emphasis on the underlying algebraic structure and on the existence of invariants. We illustrate the results with the generalized Fibonacci chains and show that the…

Zeckendorf's theorem states that every positive integer can be uniquely decomposed as a sum of nonconsecutive Fibonacci numbers. The distribution of the number of summands converges to a Gaussian, and the individual measures on gaps between…

Recent results about sums of cubes of Fibonacci numbers [Frontczak, 2018] are extended to arbitrary powers.

Let (F_n^{(k)})_{n\geq -(k-2)} be the k-generalized Fibonacci sequence, defined as the linear recurrence sequence whose first k terms are \(0, 0, \ldots, 0, 1\), and whose subsequent terms are determined by the sum of the preceding k terms.…

A multivariate Gauss-Lucas theorem is proved, sharpening and generalizing previous results on this topic. The theorem is stated in terms of a seemingly new notion of convexity. Applications to multivariate stable polynomials are given.

Zeckendorf's theorem states every positive integer has a unique decomposition as a sum of non-adjacent Fibonacci numbers. This result has been generalized to many sequences $\{a_n\}$ arising from an integer positive linear recurrence, each…

This note presents variations on the Fibonacci universal code, that may also be called the Gopala-Hemachandra code, that can have applications in source coding as well as in cryptography.

Applying the $q$-Zeilberger algorithm, we establish a unified $q$-analogue of the (C.2) and (G.2) supercongruences of Van Hamme, which can be viewed as a refinement of several previously known results. As consequences, we obtain a…

Two new generalized Fibonacci number summation identities are stated and proved, and two other new generalized Fibonacci number summation identities are derived from these, of which two special cases are already known in literature.

In this work, we define a more general family of polynomials in several variables satisfying a linear recurrence relation. Then we provide explicit formulas and determinantal expressions. Finally, we apply these results to recurrent…

We derive generalizations of a couple of inverse tangent summation identities involving Fibonacci and Lucas numbers. As byproducts we establish many new inverse tangent identities involving the Fibonacci and Lucas numbers.

By using the generalized Bernoulli numbers, we deduce new integral representations for the Riemann zeta function at positive odd-integer arguments. The explicit expressions enable us to obtain criteria for the dimension of the vector space…