On the Characteristic Polynomial of the Generalized k-Distance Tribonacci Sequences

Abstract

In 2008, I. Włoch introduced a new generalization of Pell numbers. She used special initial conditions so that this sequence describes the total number of special families of subsets of the set of n integers. In this paper, we prove some results about the roots of the characteristic polynomial of this sequence, but we will consider general initial conditions. Since there are currently several types of generalizations of the Pell sequence, it is very difficult for anyone to realize what type of sequence an author really means. Thus, we will call this sequence the generalized k-distance Tribonacci sequence (Tn(k))n≥0.

On Coding by (2,q)-Distance Fibonacci Numbers

“…This limit was probably firstly studied by Johannes Kepler in 1619 (see English translation [3]) as he formulated there the approximation of the golden ratio α by the proportions of consecutive Fibonacci numbers. There are many types of generalizations of the Fibonacci numbers (see [4][5][6][7][8][9][10][11][12][13]); e.g., changing their recurrence to the form P n = P n−2 + P n−3 , for n ≥ 3, with initial conditions P 0 = P 1 = P 2 = 1 we get Padovan numbers, which are named after R. Padovan, but in 1991 he attributed their discovery to Dutch architect Hans van der Laan and the sequences with the same recurrence were studied in 1899 by R. Perrin and in 1924 by Cordonnier (see [14][15][16]).…”