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The relations between bi-periodic jacobsthal and bi-periodic jacobsthal lucas sequence

Year 2021, Volume: 42 Issue: 2, 346 - 357, 30.06.2021

Şükran Uygun

https://doi.org/10.17776/csj.770080
Cited By: 1

Abstract

In this paper, one of the special integer sequences, Jacobsthal and Jacobsthal Lucas sequences which are encountered in computer science is generalized according to parity of the index of the entries of the sequences, called bi-periodic Jacobsthal and Jacobsthal Lucas sequences. The definitions of the bi-periodic Jacobsthal and Jacobsthal Lucas sequences are given by using classic Jacobsthal and Jacobsthal Lucas sequences. In literature, there were some relations for the bi-periodic Jacobsthal and Jacobsthal Lucas sequences. We find new identities for these sequences. If we substitute a=b=1 in the results, we get identities for classic Jacobsthal and Jacobsthal Lucas sequences.

Keywords

Bi-periodic Jacobsthal sequence Generalized Jacobsthal Lucas sequence Binet formula

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Project Number

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References

  • [1] Horadam A. F., Jacobsthal Representation Numbers, The Fibonacci Quarterly, 37 (2) (1996) 40-54.
  • [2] Edson M, Yayenie O., A New Generalization of Fibonacci Sequences and Extended Binet's Formula, Integers, 9 (2009) 639-654.
  • [3] Yayenie O., A Note on Generalized Fibonacci Sequence, Applied Mathematics and Computation, 217 (2011) 5603-5611.
  • [4] Jun S.P, Choi K.H., Some Properties of the Generalized Fibonacci Sequence {qn} by Matrix Methods, The Korean Journal of Mathematics, 24 (4) (2016) 681-691.
  • [5] Bilgici G, Two Generalizations of Lucas Sequence, Applied Mathematics and Computation, 245 (2014) 526-538.
  • [6] Uygun S., Owusu E., A New Generalization of Jacobsthal Numbers (Bi-Periodic Jacobsthal Sequences), Journal of Mathematical Analysis, 5 (2016) 728-39.
  • [7] Uygun S., Karatas H., Akıncı E., Relations on Bi-periodic Jacobsthal Sequence, Transylvanian Journal of Mathematics and Mechanics, 10 (2) (2018) 141-151.
  • [8] Uygun S., Owusu E., A Note on bi-periodic Jacobsthal Lucas Numbers, Journal of Advances in Mathematics and Computer Science, 34 (5) (2019) 1-13.
  • [9] Uygun S., Karatas H., A New Generalization of Pell-Lucas Numbers (Bi-Periodic Pell-Lucas Sequence), Communications in Mathematics and Applications, 10 (3) (2019) 1-12.
  • [10] Choo Y., Some Identities on Generalized Bi-periodic Fibonacci Sequences, International Journal of Mathematical Analysis, 13 (6) (2019) 259-267.
  • [11] Gul K., On Bi-periodic Jacobsthal and Jacobsthal-Lucas Quaternions, Journal of Mathematics Research, 11 (2) (2019) 44-52.
  • [12] Komatsu T., Ramírez J.L., Convolutions of the Bi-periodic Fibonacci Numbers, Hacettepe Journal of Mathematics & Statistics, (2019) Early Access: 1-13.
  • [13] Brigham R., Chinn P., Grimaldi R., Tiling and Patterns of Enumeration, Congressus Numerantium, 137 (1999) 207-219.
  • [14] Frey D., Sellers J., Jacobsthal Numbers and Alternating Sign Matrices, Journal. of Integer Sequences, 3 (2000).
  • [15] Grimaldi R., Binary Strings and the Jacobsthal Numbers, Congressus Numerantium, 174 (2005) 3-22.
  • [16] Grimaldi R., The Distribution of 1's in Jacobsthal Binary Sequences, Congressus Numerantium, 190 (2008) 47-64.

Year 2021, Volume: 42 Issue: 2, 346 - 357, 30.06.2021

Şükran Uygun

https://doi.org/10.17776/csj.770080
Cited By: 1

Abstract

Project Number

-

References

  • [1] Horadam A. F., Jacobsthal Representation Numbers, The Fibonacci Quarterly, 37 (2) (1996) 40-54.
  • [2] Edson M, Yayenie O., A New Generalization of Fibonacci Sequences and Extended Binet's Formula, Integers, 9 (2009) 639-654.
  • [3] Yayenie O., A Note on Generalized Fibonacci Sequence, Applied Mathematics and Computation, 217 (2011) 5603-5611.
  • [4] Jun S.P, Choi K.H., Some Properties of the Generalized Fibonacci Sequence {qn} by Matrix Methods, The Korean Journal of Mathematics, 24 (4) (2016) 681-691.
  • [5] Bilgici G, Two Generalizations of Lucas Sequence, Applied Mathematics and Computation, 245 (2014) 526-538.
  • [6] Uygun S., Owusu E., A New Generalization of Jacobsthal Numbers (Bi-Periodic Jacobsthal Sequences), Journal of Mathematical Analysis, 5 (2016) 728-39.
  • [7] Uygun S., Karatas H., Akıncı E., Relations on Bi-periodic Jacobsthal Sequence, Transylvanian Journal of Mathematics and Mechanics, 10 (2) (2018) 141-151.
  • [8] Uygun S., Owusu E., A Note on bi-periodic Jacobsthal Lucas Numbers, Journal of Advances in Mathematics and Computer Science, 34 (5) (2019) 1-13.
  • [9] Uygun S., Karatas H., A New Generalization of Pell-Lucas Numbers (Bi-Periodic Pell-Lucas Sequence), Communications in Mathematics and Applications, 10 (3) (2019) 1-12.
  • [10] Choo Y., Some Identities on Generalized Bi-periodic Fibonacci Sequences, International Journal of Mathematical Analysis, 13 (6) (2019) 259-267.
  • [11] Gul K., On Bi-periodic Jacobsthal and Jacobsthal-Lucas Quaternions, Journal of Mathematics Research, 11 (2) (2019) 44-52.
  • [12] Komatsu T., Ramírez J.L., Convolutions of the Bi-periodic Fibonacci Numbers, Hacettepe Journal of Mathematics & Statistics, (2019) Early Access: 1-13.
  • [13] Brigham R., Chinn P., Grimaldi R., Tiling and Patterns of Enumeration, Congressus Numerantium, 137 (1999) 207-219.
  • [14] Frey D., Sellers J., Jacobsthal Numbers and Alternating Sign Matrices, Journal. of Integer Sequences, 3 (2000).
  • [15] Grimaldi R., Binary Strings and the Jacobsthal Numbers, Congressus Numerantium, 174 (2005) 3-22.
  • [16] Grimaldi R., The Distribution of 1's in Jacobsthal Binary Sequences, Congressus Numerantium, 190 (2008) 47-64.
There are 16 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Şükran Uygun 0000-0002-7878-2175

Project Number -
Publication Date June 30, 2021
Submission Date July 16, 2020
Acceptance Date June 2, 2021
Published in Issue Year 2021Volume: 42 Issue: 2

Cite

APA Uygun, Ş. (2021). The relations between bi-periodic jacobsthal and bi-periodic jacobsthal lucas sequence. Cumhuriyet Science Journal, 42(2), 346-357. https://doi.org/10.17776/csj.770080

Cited By

On the Jacobsthal numbers which are the product of two Modified Pell numbers
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics
https://doi.org/10.31801/cfsuasmas.1315051