Netted Binomial Matrices

This paper describes a class of sequences that are in many ways similar to Fibonacci sequences: given n, sum the previous two terms and divide them by the largest possible power of n. The behavior of such sequences depends on n. We analyze these sequences for small n: 2, 3, 4, and 5. Surprisingly these behaviors are very different. We also talk about any n. Many statements about these sequences are difficult or impossible to prove, but they can be supported by probabilistic a...

Our aim in writing this paper is to answer to both V. E. Hoggatt, JR \cite{hogg} and Wessner\cite{wess} on the next question: find $\sum_{k=0}^n\binom{n}{k}F_{[k]}^p$, for the case $p\equiv 1\, mod\, 4$ and $p\equiv 3\, mod\, 4$. \par The case $p\equiv 0\, mod \,4$ and $p\equiv 2\, mod\, 4$, Wessner has given an answer. In particular, we give another presentation, another proof of the paper of Wessner. Our method use, essentially, the paper of Boyadzhiev\cite{boy}

In this paper we construct two types of Hessenberg matrices with the properties that every weighted isobaric polynomial (WIP) appears as a determinant of one of them, and as the permanent of the other. Every integer sequence which is linearly recurrent is representable by (an evaluation of) some linearly recurrent sequence of WIPs. WIPs are symmetric polynomials written on the elementary symmetric polynomial basis. Among them are the generalized Fibonacci polynomials and the ...

In this paper, we compute the spectral norms of the matrices related with integer squences and we give two examples related with Fibonacci and Lucas numbers.

We investigate some arithmetic properties of the q-Fibonacci numbers and the q-Pell numbers.

A sequence of nonzero integers $f = (f_1, f_2, \dots)$ is ``binomid'' if every $f$-binomid coefficient $\left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f$ is an integer. Those terms are the generalized binomial coefficients: \[ \left[\! \begin{array}{c} n \\ k \end{array}\! \right]_f \ = \ \frac{ f_nf_{n-1}\cdots f_{n-k+1} }{ f_kf_{k-1}\cdots f_1 }. \] Let $\Delta(f)$ be the infinite triangle with those numbers as entries. When $I = (1, 2, 3, \dots)$ then $\Delta(I)$ i...

In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas sequences. While the proofs of general statements rely on the binomial theorem, we prove some spacial cases by the known Pell identities. We also give extensions of obtained expressions to the other recursive sequences.

In this paper, firstly, we introduce the Q_{l}-Generating matrix for the bi-periodic Lucas numbers. Then, by taking into account this matrix representation, we obtain some properties for the bi-periodic Fibonacci and Lucas numbers.

We present an elementary proof of the generalization of the $k$-bonacci Binet formula, a closed form calculation of the $k$-bonacci numbers using the roots of the characteristic polynomial of the $k$-bonacci recursion.

In this work, we made a generalization that includes all bicomplex Fibonacci-like numbers such as; Fibonacci, Lucas, Pell, etc.. We named this generalization as bicomplex Horadam numbers. For bicomplex Fibonacci and Lucas numbers we gave some additional identities. Moreover, we have obtained the Binet formula and generating function for bicomplex Horadam numbers for the first time. We have also obtained two important identities that relate the matrix theory to the second orde...