On Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers

Abstract

In this paper, we present Binet's formulas, generating functions, and the summation formulas for generalized Pentanacci numbers, and as special cases, we investigate Pentanacci and Pentanacci-Lucas numbers with their properties. Also, we define Gaussian generalized Pentanacci numbers and as special cases, we investigate Gaussian Pentanacci and Gaussian Pentanacci-Lucas numbers with their properties. Moreover, we give some identities for these numbers. Furthermore, we present matrix formulations of generalized Pentanacci numbers and Gaussian generalized Pentanacci numbers.

Linear Summing Formulas of Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers

“…For a different proof of (a),(b),(c) see [4]. As special cases of above Theorem, we have the following two Corollaries.…”

“…for n = 1, 2, 3, .... Therefore, recurrence (1.1) holds for all integer n. Pentanacci sequence has been studied by many authors, see for example [1], [2], [3], [4].…”

A Study on Sum Formulas of Generalized Hexanacci Numbers: Closed Forms of the Sum Formulas ∑n k=0 xkWk and ∑nk=1 xkW−k

“…Name of sequence Papers which deal with summing formulas Pell and Pell-Lucas [10,11,12], [13,14] Generalized Fibonacci [15,16,17,18,19,20,21] Generalized Tribonacci [22,23,24] Generalized Tetranacci [25,26,27] Generalized Pentanacci [28,29] Generalized Hexanacci [30,31] In this work, we investigate summation formulas of generalized Hexanacci numbers.…”

On Generalized Hexanacci and Gaussian Generalized Hexanacci Numbers

“…Pethe [17] defined the complex Tribonacci numbers at Gaussian integers, see [7]. There are other several studies dedicated to these sequences of Gaussian numbers such as the works in [1,3,4,[8][9][10]12,13,15,16,[21][22][23][25][26][27], among others.…”