Tribonacci and Tribonacci-Lucas numbers via the determinants of special matrices

Abstract

In this paper, by using determinants of special matrices, it has been mainly obtained Tribonacci and Tribonacci-Lucas numbers.

On some Properties of Tribonacci Quaternions

“…For n ≥ 2, let A n = T −n and B n = K −n . Then Tribonacci and Tribonacci-Lucas sequences with negative indices are defined by the following equations, see [17];…”

“…For r = s = t = 1 and V 0 = 0, V 1 = 1, V 2 = 1, the sequence {V n } n≥0 is the well-known Tribonacci sequence denoted by {T n } n , see [2,3,12]. For r = s = t = 1 and V 0 = 3, V 1 = 1, V 2 = 3, we obtain the Tribonacci-Lucas sequence {K n } n , see [17]. The first few Tribonacci numbers and Tribonacci Lucas numbers are given in the following table: The function f (x) = a 0 + a 1 x + a 2 x 2 + · · · + a n x n + · · · is called the generating function for the sequence {a 0 , a 1 , a 2 , .…”

Sums of Tribonacci and Tribonacci-Lucas numbers

“…The properties of these numbers are studied in many articles. The most recent articles include but are not limited to [1], [2], [3], [4], [7], [8] and [9]. The Binet formulas are given by…”

“…In what follows, we will also need to define the Tribonacci numbers and the Tribonacci-Lucas numbers for negative indices. This can be done as follows (see [9]): We set T −n = B n and K −n = C n where…”

Incomplete Tribonacci–Lucas Numbers and Polynomials

“…One of these directions goes through to the Tribonacci and the Tribonacci-Lucas numbers. In fact Tribonacci numbers have been firstly defined by M. Feinberg in 1963 and then some important properties over this numbers have been created by [5,8,10,13,19]. On the other hand, Tribonacci-Lucas numbers have been introduced and investigated by author in [4].…”