In the present paper, for a positive integer $r$, we study bi-periodic $r$-Fibonacci sequence and its family of companion sequences, bi-periodic $r$-Lucas sequence of type $s$ with $1 \leq s \leq r$, which extend the classical Fibonacci and Lucas sequences. Afterwards, we establish the link between the bi-periodic $r$-Fibonacci sequence and its companion sequences. Furthermore, we give their properties as linear recurrence relations, generating functions, explicit formulas and Binet forms.
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[6] M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extended
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[7] D. Kalman, Generalized Fibonacci Numbers by matrix methods, Fibonacci Quart. 20
(1), 73-76, 1982.
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371-380, 2015.
[11] O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217,
5603-5611, 2011.
[12] Y. Yazlik, C. Köme and V. Madhusudanan, A new generalization of Fibonacci and
Lucas p-numbers, J. Comput. Anal. Appl. 25 (4), 657-669, 2018.
Year 2022,
Volume: 51 Issue: 3, 680 - 699, 01.06.2022
[1] S. Abbad, H. Belbachir and B. Benzaghou, Companion sequences associated to the
r-Fibonacci sequence: algebraic and combinatorial properties, Turk. J. Math. 43 (3),
1095-1114, 2019.
[2] H. Belbachir, A combinatorial contribution to the multinomial Chu-Vandermonde
convolution, Les Annales RECITS 1, 27-32, 2014.
[3] H. Belbachir and F. Bencherif, Linear recurrent sequences and powers of a square
matrix, Integers 6, A12, 2006.
[4] G. Bilgici, Two generalizations of Lucas sequence, Appl. Math. Comput. 245, 526-
538, 2014.
[5] L. Cerlienco, M. Mignotte and F. Piras, Suites récurrentes linéaires, propriétés algébriques et arithmétiques, Enseignement Mathématiques 33, 67-108, 1987.
[6] M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and extended
Binet’s Formula, Integers, 9, 639-654, 2009.
[7] D. Kalman, Generalized Fibonacci Numbers by matrix methods, Fibonacci Quart. 20
(1), 73-76, 1982.
[8] J.A. Raab A generalization of the connection between the Fibonacci sequence and
Pascal’s triangle, Fibonacci Quart. 1, 21-31, 1963.
[9] M. Sahin, The Gelin-Cesàro identity in some conditional sequences, Hacet. J. Math.
Stat. 40 (6), 855-861, 2011.
[10] E. Tan and A.B. Ekin, Bi-periodic Incomplete Lucas Sequences, Ars Combin. 123,
371-380, 2015.
[11] O. Yayenie, A note on generalized Fibonacci sequence, Appl. Math. Comput. 217,
5603-5611, 2011.
[12] Y. Yazlik, C. Köme and V. Madhusudanan, A new generalization of Fibonacci and
Lucas p-numbers, J. Comput. Anal. Appl. 25 (4), 657-669, 2018.
Aıt-amrane, N. R., & Belbachir, H. (2022). Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$. Hacettepe Journal of Mathematics and Statistics, 51(3), 680-699. https://doi.org/10.15672/hujms.825908
AMA
Aıt-amrane NR, Belbachir H. Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$. Hacettepe Journal of Mathematics and Statistics. June 2022;51(3):680-699. doi:10.15672/hujms.825908
Chicago
Aıt-amrane, Nacima Rosa, and Hacène Belbachir. “Bi-Periodic $r$-Fibonacci Sequence and Bi-Periodic $r$-Lucas Sequence of Type $s$”. Hacettepe Journal of Mathematics and Statistics 51, no. 3 (June 2022): 680-99. https://doi.org/10.15672/hujms.825908.
EndNote
Aıt-amrane NR, Belbachir H (June 1, 2022) Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$. Hacettepe Journal of Mathematics and Statistics 51 3 680–699.
IEEE
N. R. Aıt-amrane and H. Belbachir, “Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, pp. 680–699, 2022, doi: 10.15672/hujms.825908.
ISNAD
Aıt-amrane, Nacima Rosa - Belbachir, Hacène. “Bi-Periodic $r$-Fibonacci Sequence and Bi-Periodic $r$-Lucas Sequence of Type $s$”. Hacettepe Journal of Mathematics and Statistics 51/3 (June 2022), 680-699. https://doi.org/10.15672/hujms.825908.
JAMA
Aıt-amrane NR, Belbachir H. Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$. Hacettepe Journal of Mathematics and Statistics. 2022;51:680–699.
MLA
Aıt-amrane, Nacima Rosa and Hacène Belbachir. “Bi-Periodic $r$-Fibonacci Sequence and Bi-Periodic $r$-Lucas Sequence of Type $s$”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 3, 2022, pp. 680-99, doi:10.15672/hujms.825908.
Vancouver
Aıt-amrane NR, Belbachir H. Bi-periodic $r$-Fibonacci sequence and bi-periodic $r$-Lucas sequence of type $s$. Hacettepe Journal of Mathematics and Statistics. 2022;51(3):680-99.