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The Fibonacci sequence is a sequence of numbers that has been studied for hundreds of years. In this paper, we introduce the new sequence S_{k,n} with initial conditions S_{k,0} = 2b and S_{k,1} = bk + a, which is generated by the recurrence relation S_{k,n} = kS_{k,n-1} +S{k,n-2} for n >= 2, where a, b, k are real numbers. Using the sequence S_{k,n}, we introduce and prove some special identities. Also, we deal with the circulant and skew circulant matrices for the sequence S_{k,n}.
This study applies the binomial, k-binomial, rising k-binomial and falling k-binomial transforms to the modified k-Fibonacci-like sequence. Also, the Binet formulas and generating functions of the above mentioned four transforms are newly found by the recurrence relations.
We introduce a generalized $k$-FL sequence and special kind of pairs of real numbers that are related to it, and give an application on the integral solutions of a certain equation using those pairs. Also, we associate skew circulant and circulant ma
The generalized Fibonacci sequences are sequences ${f_n}$ which satisfy the recurrence $f_n(s, t) = sf_{n - 1}(s, t) + tf_{n - 2}(s, t)$ ($s, t in mathbb{Z}$) with initial conditions $f_0(s, t) = 0$ and $f_1(s, t) = 1$. In a recent paper, Amdeberhan,
We study higher-dimensional interlacing Fibonacci sequences, generated via both Chebyshev type functions and $m$-dimensional recurrence relations. For each integer $m$, there exist both rational and integ
In 1997, Bousquet-Melou and Eriksson stated a broad generalization of Eulers distinct-odd partition theorem, namely the $(k,l)$-Euler theorem. Their identity involved the $(k,l)$-lecture-hall partitions, which, unlike usual difference conditions of p