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This paper focuses on a specially constructed matrix whose entries are harmonic Fibonacci numbers and considers its Hadamard exponential matrix. A lot of admiring algebraic properties are presented for both of them. Some of them are determinant, inverse in usual and in the Hadamard sense, permanents, some norms, etc. Additionally, a MATLAB-R2016a code is given to facilitate the calculations and to further enrich the content.

At this work, we give a method for constructing the Perrin and Padovan sequences and obtain the De Moivre-type identity for Padovan numbers. Also, we define a Padovan sequence with new initial conditions and find some identities between all of these auxiliary sequences. Furthermore, we give quadratic approximations for these sequences.

The main aim of this study is to obtain De Moivre-type identities for Jacobsthal numbers. Also, this paper presents a method for constructing the second order Jacobsthal and Jacobsthal third-order numbers and the third-order Jacobsthal and Jacobsthal–Lucas numbers. Moreover, we give some interesting identities, such as Binet’s formulas for some specific third-order Jacobsthal numbers that we derive from De Moivre-type identities.

This paper aims to present a method for constructing the second order Pell and Pell-Lucas numbers and the third order Pell and Pell-Lucas numbers. Moreover, we obtain the De Moivre-type identities for these numbers. In addition, we define a Pell sequence with new initial conditions and give some identities for these third order Pell numbers such as Binet's formulas, generating functions, sums.