Fibonacci-Like sequence associated with k-Pell, k-Pell-Lucas and Modified k-Pell sequences

Abstract

Abstract. Fibonacci sequence is a well known example of second order linear recurrence relatio. Besides Fibonacci numbers and their generalizations have many applications as well as interesting properties almost in every field of science such as in Physics, Biology, Mathematics (Algebra, Geometry and Number Theory itself). The main aim of the present article to introduce a generalization of Fibonacci sequence which is similar to k-Pell, k-Pell-Lucas, Modified k-Pell sequences and known as Fibonacci-Like sequence. After that we obtain some fundamental properties of Fibonacci-Like sequence such as Binet formulae of Fibonacci-Like sequence, binomial transform of the Fibonacci-Like sequence and sum of Fibonacci-Like numbers with indexes in an arithmetic sequence. In addition to this we obtain some new relations among k-Pell, k-Pell-Lucas, Modified k-Pell and Fibonacci-Like sequences.

Matrix representations of Fibonacci-like sequences

“…with R k,0 = 2 and R k,1 = 1 ( [15]). Clearly x 2 − 2x − k = 0 is also the characteristic equation of R k,n = 2R k,n−1 + kR k,n−2 recurrence formula in equation 5and a and b are its two roots where a = 1 + √ 1 + k and b = 1 − √ 1 + k. Many properties of these number sequences are deduced directly from elementary matrix algebra.…”