Revealing Hidden Patterns Powers of Jacobsthal, Pell and Mersenne Sequences via the Powers of Lower Terms and Some Applications

Mirwais Mansoor Kakar

doi:10.5644/SJM.22.01.02

Authors

Mirwais Mansoor Kakar Ghazni University, Department of Mathematics, Ghazni, 2301, Afghanistan

DOI:

https://doi.org/10.5644/SJM.22.01.02

Keywords:

Jacobsthal numbers, Pell numbers, Mersenne numbers, Power furmulas, Recurrence relations, Generating functions

Abstract

In this paper, we investigate the powers of Jacobsthal, Pell, and Mersenne numbers expressed as the sum of the powers of lower terms. Specifically, we derive formulas for the $m^{th}$ power of $\mathrm{A}_{n+1} \in \{\mathrm{J}_{n+1}, \mathrm{P}_{n+1}, \mathrm{M}_{n+1}\}$ in terms of the sum of the $m^{th}$ powers of the lower terms $\mathrm{A}_n, \mathrm{A}_{n-1}, \dots, \mathrm{A}_{n-m}$.
We also present some applications of these powers to determine recurrence relations for the integer sequences of the squares and cubes of Jacobsthal, Pell, and Mersenne numbers. Additionally, a few numerical examples are provided for each case.

Statistics

Abstract: 3 / PDF: 0

References

M. Bicknell, A Primer of the Pell Sequence and Related Sequences, Fibonacci Quart., 13(4) (1975), 345–349.

A. Boussayoud, M. Chelgham, and S. Boughaba, On Some Identities and Generating Functions for Mersenne Numbers and Polynomials, Turkish J. Anal. Number Theory, 6(3) (2018), 93–97.

B. A. Brousseau, A Sequence of Power Formulas, Fibonacci Quart., 6(1) (1968), 81–83.

P. Catarino et al., New Families of Jacobsthal and Jacobsthal–Lucas Numbers, Algebra Discret. Math., 20 (2015), 40–54.

P. Catarino, H. Campos, and P. Vasco, On the Mersenne Sequence, Annales Mathematicae et Informaticae, 46 (2016), 37–53.

A. Dasdemir, Mersenne, Jacobsthal, and Jacobsthal–Lucas Numbers with Negative Subscripts, Acta Math. Univ. Comenianae, 88(1) (2019), 145–156.

T. Goy, On New Identities for Mersenne Numbers, Applied Mathematics E-Notes, 18 (2018), 100–105.

A. F. Horadam, Jacobsthal Representation Numbers, Fibonacci Quart., 34 (1996), 40–54.

A. F. Horadam, Jacobsthal Representation Polynomials, Fibonacci Quart., 35 (1997), 137–148.

A. F. Horadam, Jacobsthal and Pell Curves, Fibonacci Quart., 26 (1988), 79–83.

A. F. Horadam, Pell Identities, Fibonacci Quart., 9(3) (1961), 245–252.

T. Koshy, Pell and Pell–Lucas Numbers with Applications, Springer, 2014.

T. Koshy, Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, 2001.

R. Melham, Sums Involving Fibonacci and Pell Numbers, Portugaliae Math., 56(3) (1999), 309–317.

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, 1964.

Y. Soykan, E. Taşdemir, and M. Gocen, Binomial transform of the generalized third-order Jacobsthal sequence, Asian-European Journal of Mathematics, 15(12) (2022).

Y. Soykan and E. Taşdemir, A note on k-circulant matrices with the generalized third-order Jacobsthal numbers, Advanced Studies: Euro-Tbilisi Mathematical Journal, 17(4) (2024), 81–101.

Y. Soykan and E. E. Polatlı, A note on binomial transform of the generalized fifth order Jacobsthal numbers, Open Journal of Discrete Applied Mathematics, 5 (2022), 1–24.

Y. Soykan, I. Okumuş, and E. Taşdemir, On Generalized Tribonacci Sedenions, Sarajevo Journal of Mathematics, 16(1) (2020), 103–122.