A family of lacunary recurrences for Lucas Numbers

Analogues of a Fibonacci-Lucas Identity

Sury's 2014 proof of an identity for Fibonacci and Lucas numbers (Identity 236 of Benjamin and Quinn's 2003 book: {\em Proofs that count: The art of combinatorial proof}) has excited a lot of comment. We give an alternate, telescoping, proof of this---and associated---identities and generalize them. We also give analogous identities for other sequences that satisfy a three-term recurrence relation.

On Horadam-Lucas sequence

Horadam introduced a new generalized sequence of numbers, describing its key features and the special sub-sequences that are obtained depending on the choices of initial parameters. This sequence and its sub-sequences are known as the Horadam, generalized Fibonacci, and generalized Lucas numbers, respectively. In the present study, we propose another new sequence, which satisfies a second-order recurrence relation, in addition to Horadam's definition. Further, we prove the Binet's formula, some famous identities, and summation formulas for this new sequence. In particular, we demonstrate the i…

斐波那契-富比尼数与卢卡斯-富比尼数

The Fibonacci-Fubini and Lucas-Fubini numbers

本文通过将有序贝尔数中的划分块按斐波那契方式排列，引入斐波那契-富比尼数，并研究其生成函数、显式公式、组合解释及算术三角形，同时揭示其与斯特林数的联系。

Divisors and specializations of Lucas polynomials

Three-term recurrences have infused stupendous amount of research in a broad spectrum of the sciences, such as orthogonal polynomials (in special functions) and lattice paths (in enumerative combinatorics). Among these are the Lucas polynomials, which have seen a recent true revival. In this paper one of the themes of investigation is the specialization to the Pell and Delannoy numbers. The underpinning motivation comprises primarily of divisibility and symmetry. One of the most remarkable findings is a structural decomposition of the Lucas polynomials into what we term as flat and sharp analo…

新巴尔科平衡数

Neo balcobalancing numbers

本文定义了新巴尔科平衡数等四类新数，推导其通项公式及与传统数论序列的关系、前n项和公式，同时实现了新旧数类通项与求和公式的互推。

关于由k-斐波那契数和k-卢卡斯数乘积定义的矩阵的一些性质

On Some Properties of Matrices with Entries Defined by Products of $k$-Fibonacci and $k$-Lucas Numbers

本文研究了由k-斐波那契数和k-卢卡斯数乘积定义的矩阵家族，推导了多个矩阵不变量的显式公式，并探讨了与整数序列的关系。

Generalized Fibonacci recurrences and the lex-least De Bruijn sequence

The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographically-least de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and Lucas numbers, respectively.

Relation between powers of factors and recurrence function characterizing Sturmian words

In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.

Appearance of Balancing and related number sequences in steady state probabilities of some Markov chains

Balancing and Lucas-balancing numbers are solutions of a Diophantine equation and satisfy a second order homogeneous recurrence relation. Interestingly, these numbers can be seen as numerators and denominators in the steady state probabilities of a class of transition probability matrices of Markov chains. An identity relating the balancing numbers and the silver ratio can be obtained as a byproduct.

GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

We explore the sums of $k$ consecutive terms in the generalized Fibonacci sequence $\left(G_n\right)_{n \geq 0}$ given by the recurrence $G_n = G_{n-1} + G_{n-2}$ for all $n \geq 2$ with integral initial conditions $G_0$ and $G_1$. In particular, we give precise values for the greatest common divisor (GCD) of all sums of $k$ consecutive terms of $\left(G_n\right)_{n \geq 0}$. When $G_0 = 0$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Fibonacci numbers, and when $G_0 = 2$ and $G_1 = 1$, we yield the GCD of all sums of $k$ consecutive Lucas numbers. Denoting the GCD of all sum…

On the properties of new families of generalized Fibonacci numbers

In this paper, new families of generalized Fibonacci and Lucas numbers are introduced. In addition, we present the recurrence relations and the generating functions of the new families for $k=2$.

Polymatrix and generalized polynacci numbers

We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the definition of reflected sequence and inverted sequence and we establish some relationship between the coefficients of the Cayley-Hamilton equation for these matrices and the introduced sequences.

New Tribonacci Recurrence Relations and Addition Formulas

Only one three-term recurrence relation, namely, $W_{r}=2W_{r-1}-W_{r-4}$, is known for the generalized Tribonacci numbers, $W_r$, $r\in\mathbb{Z}$, defined by $W_{r}=W_{r-1}+W_{r-2}+W_{r-3}$ and \mbox{$W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}$}, where $W_0$, $W_1$ and $W_2$ are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is, $W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s-2} )W_r + T_s W_{r + 1}$, where $({T_r})_{r\in\mathbb{Z}}$ is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with $W_0=T…

Power law behavior associated with a Fibonacci Lucas model and generalized statistical models

A Fibonacci-Lucas based statistical model and several other related models are studied. The canonical and grand canonical partition functions for these models are developed.Partition structure such as the distribution of sizes as in a cluster distribution is explored.Ensemble averaging over all partitions leads to a scale invariant power law behavior at a particular critical like point. The canonical ensemble of the Fibonacci-Lucas case involves the Gegenbauer polynomial.The model has a hyperbolic power law behavior, a feature linked to the golden mean ratio of two adjacent Fibonacci numbers a…