On the Bi-Periodic Mersenne Sequence

In this paper, the bi-periodic Mersenne sequence, which is a generalization of the Mersenne sequence, is defined. The characteristic function, generating function and Binet’s formula for this sequence are obtained. Also, by using Binet’s formula, some important identities and properties for the bi-periodic Mersenne sequence are presented.

Keywords:

Binet formula, Catalan identity Generating function, Mersenne numbers,

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[1] E. Özkan, A. Aydoğdu, İ. Altun, Some identities for a family of Fibonacci and Lucas numbers, J. Math. Stat. Sci., 3 (2017), 295-303.
[2] S. Çelik, İ. Durukan, E. Özkan, New recurrences on Pell numbers, Pell-Lucas numbers, Jacobsthal numbers, and Jacobsthal-Lucas numbers, Chaos Solitons Fractals,150 (2021), 111173.
[3] E. Özkan, N. Ş . Yilmaz, A. Włoch, On F3(k,n)-numbers of the Fibonacci type, Bol. Soc. Mat. Mex.,27 (2021), 77.
[4] T. Koshy, Pell and Pell–Lucas Numbers with Applications, Springer, New York, 2014.
[5] E. Özkan, M. Uysal, Mersenne-Lucas hybrid numbers, Math. Montisnigri, 52 (2021), 17-29.
[6] P. Catarino, H. Campos, P. Vasco, On the Mersenne sequence, Ann. Math. Inform., 46 (2016),37-53.
[7] T. Koshy, Fibonacci and Lucas Numbers with Applications, John Wiley and Sons Inc., New York, 2001.
[8] A. F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly, 68 (1961), 455-459.
[9] G. Özkan Kızılırmak, On some identities and Hankel matrices norms involving new defined generalized modified Pell numbers, J. New Results Sci., 10 (2021), 60-66.
[10] G. Bilgici, New generalizations of Fibonacci and Lucas numbers, Appl. Math. Sci., 8 (2014), 1429-1437.
[11] S. Falcon, A. Plaza, On the Fibonacci k-numbers, Chaos Solitions Fractals, 32 (2007), 1615-1624.
[12] A. Szynal-Liana, A. Włoch, I. Włoch, On generalized Pell numbers generated by Fibonacci and Lucas numbers, Ars Combin., 115 (2014), 411-423.
[13] N. Yılmaz, A. Aydog˘du, E. Özkan, Some properties of k-generalized Fibonacci numbers, Math. Montisnigri, 50 (2021), 73-79.
[14] P. Catarino, On some identities for k-Fibonacci sequence, Int. J. Contemp. Math. Sci., 9 (2014), 37- 42.
[15] S. P. Pethe, C. N. Phadte, A generalization of the Fibonacci sequence, Appl. Fibonacci Numbers, 5 (1992), 465-472.
[16] O. M. Yayenie, A. Edson, New generalization of Fibonacci sequences and extended Binet’s formula, Integers, 9 (2009), 639-654.
[17] D. Tasci, E. Sevgi, Bi-periodic Balancing numbers, J. Sci. Arts, 1 (2020), 75-84.
[18] S. Uygun, E. Owusu, A new generalization of Jacobsthal Lucas numbers, J. Adv. Math. Comput. Sci., 7 (2016), 28-39.
[19] J. L. Ramirez, Bi-periodic incomplete Fibonacci sequences, Ann. Math. Inform., 42 (2013), 83–92.
[20] S. Uygun, H. Karatas, A new generalization of Pell-Lucas numbers (Bi-periodic Pell-Lucas sequence), Commun. Math. Appl.,10 (2019), 469–479.