1. bookAHEAD OF PRINT
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
access type Open Access

On Generalized Jacobsthal and Jacobsthal–Lucas Numbers

Published Online: 01 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 05 Dec 2021
Accepted: 16 Jun 2022
Journal Details
License
Format
Journal
eISSN
2391-4238
First Published
01 Jan 1985
Publication timeframe
2 times per year
Languages
English
Abstract

Jacobsthal numbers and Jacobsthal–Lucas numbers are some of the most studied special integer sequences related to the Fibonacci numbers. In this study, we introduce one parameter generalizations of Jacobsthal numbers and Jacobsthal–Lucas numbers. We define two sequences, called generalized Jacobsthal sequence and generalized Jacobsthal–Lucas sequence. We give generating functions, Binet’s formulas for these numbers. Moreover, we obtain some identities, among others Catalan’s, Cassini’s identities and summation formulas for the generalized Jacobsthal numbers and the generalized Jacobsthal–Lucas numbers. These properties generalize the well-known results for classical Jacobsthal numbers and Jacobsthal–Lucas numbers. Additionally, we give a matrix representation of the presented numbers.

Keywords

MSC 2010

[1] D. Bród, On a two-parameter generalization of Jacobsthal numbers and its graph interpretation, Ann. Univ. Mariae Curie-Skłodowska Sect. A 72 (2018), no. 2, 21–28. Search in Google Scholar

[2] D. Bród, On a new Jacobsthal–type sequence, Ars Combin. 150 (2020), 21–29. Search in Google Scholar

[3] G.B. Djordjević, Some generalizations of the Jacobsthal numbers, Filomat 24 (2010), no. 2, 143–151. Search in Google Scholar

[4] S. Falcon, On the k-Jacobsthal numbers, American Review of Mathematics and Statistics 2 (2014), no. 1, 67–77. Search in Google Scholar

[5] S. Halici and M. Uysal, A study on some identities involving (sk, t)-Jacobsthal numbers, Notes Number Theory Discrete Math. 26 (2020), no. 4, 74–79. Search in Google Scholar

[6] A.F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), no. 1, 40–54. Search in Google Scholar

[7] D. Jhala, K. Sisodiya, and G.P.S. Rathore, On some identities for k-Jacobsthal numbers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 12, 551–556. Search in Google Scholar

[8] F. Köken and D. Bozkurt, On the Jacobsthal–Lucas numbers by matrix method, Int. J. Contemp. Math. Sci. 3 (2008), no. 33, 1629–1633. Search in Google Scholar

[9] N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. Avaliable at https://oeis.org/book.html. Search in Google Scholar

[10] A. Szynal-Liana, A. Włoch, and I. Włoch, On generalized Pell numbers generated by Fibonacci and Lucas numbers, Ars Combin. 115 (2014), 411–423. Search in Google Scholar

[11] A.A. Wani, P. Catarino, and S. Halici, On a study of (s, t)-generalized Pell sequence and its matrix sequence, Punjab Univ. J. Math. (Lahore) 51 (2019), no. 9, 17–32. Search in Google Scholar

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