Many quaternion numbers associated with Fibonacci and Lucas or even their generalizations have been defined widely discussed so far. In all the studies, coefficients of these quaternions selected from consecutive terms numbers. this study, we define other for usual quaternions. We also present some properties, including Binet's formulas d'Ocagne's identities, types quaternions.

This is a detailed version of my text [2]. It contains the proof outlined in [2] in much more detail and was written for the purpose of persuading myself that my proofs are correct. This note has never been proofread by myself or anyone else. If you find any mistakes or typos, please inform me at ∆Γ@gmail.com where ∆ =darij and Γ =grinberg Thank you! Definitions. 1) A subset S of Z is called ho...

for n ~ 0, Lo = 2, L1 = 1, Fo = 0 and FI = 1. These sequences playa very important role in the studies of the theory and application of mathematics. Therefore, the various properties of Ln and Fn were investigated by many authors. For example, R. L. Duncan [1] and L. Kuipers [2J proved that (logFn) is uniformly distributed mod 1. H.London and R.Finkelstein [3] studied the Fibonacci and Lucas nu...

In this paper, we define and investigate modified p-Leonardo, p-Leonardo-Lucas p-Leonardo sequences as special cases of the generalized Leonardo sequence. We present Binet's formulas, generating functions, Simson summation formulas for these sequences. Moreover, give some identities matrices related with Furthermore, show that there are close relations between p-Leonardo-Lucas, numbers Fibonacc...

In this paper, using the number of spanning trees in some classes of graphs, we prove the identities: Fn = 2n−1 n √

Many quaternions with the coefficients selected from special integer sequences such as Fibonacci and Lucas have been investigated by a great number of researchers. This article presents new classes whose components are composed symmetrical hyperbolic functions. In addition, Binet's formulas, certain generating matrices, functions, Cassini's d'Ocagne's identities for these given.

In this study, we define a generalization of Lucas sequence {pn}. Then we obtain Binet formula of sequence {pn} . Also, we investigate relationships between generalized Fibonacci and Lucas sequences.

Combinatorial proofs are appealing since they lead to intuitive understanding. Proofs based on other mathematical techniques may be convincing, but still leave the reader wondering why the result holds. A large collection of combinatorial proofs is presented in [1], including many proofs of Fibonacci identities based on counting tilings of a one-dimensional board with squares and dominoes. An a...

The Mulatu numbers were studied [1] and [2]. are sequences of the form: 4, 5,6,11,17,28,45...The have wonderful amazing properties patterns.In mathematical terms, sequence is defined by following recurrence relation: In [2] some patterns considered. this paper, we investigate additionalproperties these fascinating numbers. Many beautiful identities involving Mulatunumbers in relation with Fibon...