Research Article
Year 2021,
Volume: 42 Issue: 2, 346 - 357, 30.06.2021
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.
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Thanks
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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
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 | |
| 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 |