Generalized Olivier Numbers
Asian Research Journal of Mathematics,
Page 1-22
DOI:
10.9734/arjom/2023/v19i1634
Abstract
In this paper, we introduce and investigate the generalized Olivier sequences and we deal with, in detail, two special cases, namely, Olivier and Olivier-Lucas sequences. We present Binet's formulas, generating functions, Simson formulas, and the summation formulas for these sequences. We also provide various
matrices and identities associated with these sequences. Furthermore, we show that there are close relations between Olivier, Olivier-Lucas and adjusted Pell-Padovan, third order Lucas-Pell, third order Fibonacci-Pell, Pell-Perrin, Pell-Padovan numbers. Moreover, we give some identities and matrices related with these
sequences.
Keywords:
- Olivier numbers
- Olivier-Lucas numbers
- Pell-Padovan numbers
- Pell-Perrin numbers
- third order Fibonacci-Pell numbers
- third order Lucas-Pell numbers numbers
- Fibonacci numbers
- Lucas numbers
How to Cite
Soykan, Y. (2023). Generalized Olivier Numbers. Asian Research Journal of Mathematics, 19(1), 1-22. https://doi.org/10.9734/arjom/2023/v19i1634
References
Sloane NJA. The on-line encyclopedia of integer sequences. Available: http://oeis.org/
Soykan Y. Generalized Pell-Padovan Numbers, Asian Journal of Advanced Research and Reports.
;11(2):8-28. DOI: 10.9734/AJARR/2020/v11i230259
Hathiwala GS, Shah DV. Binet{Type Formula For The Sequence of Tetranacci Numbers by Alternate
Methods. Mathematical Journal of Interdisciplinary Sciences. 2017;6(1):37{48.
Melham RS. Some Analogs of the Identity F2
n + F2
n+1 = F2
n+1, Fibonacci Quarterly. 1999;305-311.
Natividad LR. On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order. International Journal
of Mathematics and Computing. 2013;3(2):38-40.
Singh B, Bhadouria P, Sikhwal O, Sisodiya K. A Formula for Tetranacci-Like Sequence. Gen. Math. Notes.
;20(2):136-141.
Soykan Y. Properties of Generalized (r,s,t,u)-Numbers. Earthline Journal of Mathematical Sciences.
;5(2):297-327.
Available: https://doi.org/10.34198/ejms.5221.297327
Soykan Y. A study on generalized bonacci polynomials: Sum formulas. International Journal of Advances
in Applied Mathematics and Mechanics. 2022;10(1), 39-118. (ISSN: 2347-2529)
Soykan Y. On Generalized Fibonacci Polynomials: Horadam Polynomials. Earthline Journal of
Mathematical Sciences. 2023;11(1):23-114.
E-ISSN: 2581-8147.
Available:https://doi.org/10.34198/ejms.11123.23114
Waddill ME, Sacks L. Another generalized Fibonacci sequence. The Fibonacci Quarterly. 1967;5(3):209-22..
Waddill ME. The Tetranacci sequence and generalizations. The Fibonacci Quarterly. 1992;30(1):9-20.
Howard FT, Saidak F. Zhou's theory of constructing identities. In Congress Numer. 2010;200:225-237.
Soykan Y. Simson Identity of Generalized m-step Fibonacci Numbers. International Journal of Advances
in Applied Mathematics and Mechanics. 2019;7(2):45-56.
Soykan Y. A Study On the Recurrence Properties of Generalized Tetranacci Sequence. International Journal
of Mathematics Trends and Technology. 2021;67(8):185-92.
DOI:10.14445/22315373/IJMTT-V67I8P522
Soykan Y. On the recurrence properties of generalized Tribonacci sequence. Earthline Journal of
Mathematical Sciences. 2021;6(2):253-69.
https://doi.org/10.34198/ejms.6221.253269
Abd-Elhameed WM, Youssri YH. Solutions of the connection problems between Fermat and generalized
Fibonacci polynomials. JP Journal of Algebra, Number Theory and Applications. 2016;38(4):349-362.
Available: http://dx.doi.org/10.17654/NT038040349
Koshy T, Fibonacci and Lucas Numbers with Applications. Wiley-Interscience. New York; 2001.
Marohnic L, Strmecki T. Plastic Number: Construction and Applications. Advanced Research in Scientic
Areas, 2012;3(7):1523-1528.
Padovan R. Dom hans van der Laan and the plastic number. InArchitecture and Mathematics from
Antiquity to the Future. 2015;407-419. Birkhauser, Cham.
Available: https://doi.org/10.1007/978-3-319-00143-2 27.
Padovan R, Dom Hans van der Laan: Modern Primitive, Architectura and Natura Press; 1994.
Available: http://www.nexusjournal.com/conferences/N2002-Padovan.html)
Cerda-Morales G. Identities for Third Order Jacobsthal Quaternions, Advances in Applied Cliord
Algebras. 2017;27(2):1043|1053.
Soykan; Asian Res. J. Math., vol. 19, no. 1, pp. 1-22, 2023; Article no.ARJOM.95530
Cerda-Morales G. On a Generalization of Tribonacci Quaternions, Mediterranean Journal of Mathematics.
;14:239:1-12.
Yilmaz N, Taskara N. Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices.
Applied Mathematical Sciences. 2014;8(39):1947-1955.
Shtayat J, Al-Kateeb A. An Encoding-Decoding algorithm based on Padovan numbers.
;arXiv:1907.02007.
Basu M, Das M, Tribonacci Matrices and a New Coding Theory, Discrete Mathematics, Algorithms and
Applications. 2014;6 (1)1450008:17.
Deveci O, Shannon, AG. Pell{Padovan-Circulant Sequences and Their Applications, Notes on Number
Theory and Discrete Mathematics. 2017;23(3):100{114.
Gocen M. The Exact Solutions of Some Dierence Equations Associated with Adjusted Jacobsthal-Padovan
Numbers, Krklareli University Journal of Engineering and Science. 2022;8(1):1-14.
DOI: 10.34186/klujes.1078836
Sunitha K, Sheriba M. Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs, Ratio
Mathematica. 2022;44:188-196.
Dikmen CM, Altnsoy M. On Third Order Hyperbolic Jacobsthal Numbers, Konuralp Journal of
Mathematics. 2022;10(1):118-126.
Kuloglu B, O zkan E, Shannon AG. The Narayana Sequence in Finite Groups, Fibonacci Quarterly.
;60(5):212{221.
Soykan Y, Tasdemir E, Gocen M. Binomial Transform of the Generalized Third-Order Jacobsthal Sequence,
Asian-European Journal of Mathematics. 2022;15(12).
Available: https://doi.org/10.1142/S1793557122502242.
Soykan Y, Tasdemir E, Okumus, _I. A Study on Binomial Transform of the Generalized Padovan Sequence,
Journal of Science and Arts. 2022;22(1), 63-90.
Available: https://doi.org/10.46939/J.Sci.Arts-22.1-a06
Soykan Y. On Dual Hyperbolic Generalized Fibonacci Numbers. Indian J Pure Appl Math; 2021.
Available: https://doi.org/10.1007/s13226-021-00128-2
Soykan Y, Ta sdemir E, Okumus _I, Gocen M. Gaussian Generalized Tribonacci Numbers. Journal of
Progressive Research in Mathematics(JPRM). 2018;14 (2):2373-2387.
Soykan Y, Okumus _I, Tasdemir E. On Generalized Tribonacci Sedenions, Sarajevo Journal of Mathematics,
;16(1):103-122.
ISSN 2233-1964.
DOI: 10.5644/SJM.16.01.08
Soykan Y. Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, Communications in Mathematics
and Applications. 2020;11(2):281-295.
DOI: 10.26713/cma.v11i2.1102
Soykan Y. Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with
the Generalized Tribonacci Numbers, Earthline Journal of Mathematical Sciences. 2021;6(1):131-151.
Available: https://doi.org/10.34198/ejms.6121.131151
Soykan Y, Tetranacci and Tetranacci-Lucas Quaternions, Asian Research Journal of Mathematics.
;15(1): 1-24; Article no.ARJOM.50749.
Soykan Y. Gaussian Generalized Tetranacci Numbers, Journal of Advances in Mathematics and Computer
Science. 2019;31(3):1-21, Article no.JAMCS.48063.
Soykan; Asian Res. J. Math., vol. 19, no. 1, pp. 1-22, 2023; Article no.ARJOM.95530
Soykan Y. Matrix Sequences of Tetranacci and Tetranacci-Lucas Numbers, Int. J. Adv. Appl. Math. and
Mech. 2019;7(2):57-69. (ISSN: 2347-2529).
Soykan Y. On Binomial Transform of the Generalized Tetranacci Sequence, International Journal of
Advances in Applied Mathematics and Mechanics. 2021;9(2):8-27.
Soykan Y. Generalized Pell-Padovan Numbers, Asian Journal of Advanced Research and Reports.
;11(2):8-28. DOI: 10.9734/AJARR/2020/v11i230259
Hathiwala GS, Shah DV. Binet{Type Formula For The Sequence of Tetranacci Numbers by Alternate
Methods. Mathematical Journal of Interdisciplinary Sciences. 2017;6(1):37{48.
Melham RS. Some Analogs of the Identity F2
n + F2
n+1 = F2
n+1, Fibonacci Quarterly. 1999;305-311.
Natividad LR. On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order. International Journal
of Mathematics and Computing. 2013;3(2):38-40.
Singh B, Bhadouria P, Sikhwal O, Sisodiya K. A Formula for Tetranacci-Like Sequence. Gen. Math. Notes.
;20(2):136-141.
Soykan Y. Properties of Generalized (r,s,t,u)-Numbers. Earthline Journal of Mathematical Sciences.
;5(2):297-327.
Available: https://doi.org/10.34198/ejms.5221.297327
Soykan Y. A study on generalized bonacci polynomials: Sum formulas. International Journal of Advances
in Applied Mathematics and Mechanics. 2022;10(1), 39-118. (ISSN: 2347-2529)
Soykan Y. On Generalized Fibonacci Polynomials: Horadam Polynomials. Earthline Journal of
Mathematical Sciences. 2023;11(1):23-114.
E-ISSN: 2581-8147.
Available:https://doi.org/10.34198/ejms.11123.23114
Waddill ME, Sacks L. Another generalized Fibonacci sequence. The Fibonacci Quarterly. 1967;5(3):209-22..
Waddill ME. The Tetranacci sequence and generalizations. The Fibonacci Quarterly. 1992;30(1):9-20.
Howard FT, Saidak F. Zhou's theory of constructing identities. In Congress Numer. 2010;200:225-237.
Soykan Y. Simson Identity of Generalized m-step Fibonacci Numbers. International Journal of Advances
in Applied Mathematics and Mechanics. 2019;7(2):45-56.
Soykan Y. A Study On the Recurrence Properties of Generalized Tetranacci Sequence. International Journal
of Mathematics Trends and Technology. 2021;67(8):185-92.
DOI:10.14445/22315373/IJMTT-V67I8P522
Soykan Y. On the recurrence properties of generalized Tribonacci sequence. Earthline Journal of
Mathematical Sciences. 2021;6(2):253-69.
https://doi.org/10.34198/ejms.6221.253269
Abd-Elhameed WM, Youssri YH. Solutions of the connection problems between Fermat and generalized
Fibonacci polynomials. JP Journal of Algebra, Number Theory and Applications. 2016;38(4):349-362.
Available: http://dx.doi.org/10.17654/NT038040349
Koshy T, Fibonacci and Lucas Numbers with Applications. Wiley-Interscience. New York; 2001.
Marohnic L, Strmecki T. Plastic Number: Construction and Applications. Advanced Research in Scientic
Areas, 2012;3(7):1523-1528.
Padovan R. Dom hans van der Laan and the plastic number. InArchitecture and Mathematics from
Antiquity to the Future. 2015;407-419. Birkhauser, Cham.
Available: https://doi.org/10.1007/978-3-319-00143-2 27.
Padovan R, Dom Hans van der Laan: Modern Primitive, Architectura and Natura Press; 1994.
Available: http://www.nexusjournal.com/conferences/N2002-Padovan.html)
Cerda-Morales G. Identities for Third Order Jacobsthal Quaternions, Advances in Applied Cliord
Algebras. 2017;27(2):1043|1053.
Soykan; Asian Res. J. Math., vol. 19, no. 1, pp. 1-22, 2023; Article no.ARJOM.95530
Cerda-Morales G. On a Generalization of Tribonacci Quaternions, Mediterranean Journal of Mathematics.
;14:239:1-12.
Yilmaz N, Taskara N. Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices.
Applied Mathematical Sciences. 2014;8(39):1947-1955.
Shtayat J, Al-Kateeb A. An Encoding-Decoding algorithm based on Padovan numbers.
;arXiv:1907.02007.
Basu M, Das M, Tribonacci Matrices and a New Coding Theory, Discrete Mathematics, Algorithms and
Applications. 2014;6 (1)1450008:17.
Deveci O, Shannon, AG. Pell{Padovan-Circulant Sequences and Their Applications, Notes on Number
Theory and Discrete Mathematics. 2017;23(3):100{114.
Gocen M. The Exact Solutions of Some Dierence Equations Associated with Adjusted Jacobsthal-Padovan
Numbers, Krklareli University Journal of Engineering and Science. 2022;8(1):1-14.
DOI: 10.34186/klujes.1078836
Sunitha K, Sheriba M. Gaussian Tribonacci R-Graceful Labeling of Some Tree Related Graphs, Ratio
Mathematica. 2022;44:188-196.
Dikmen CM, Altnsoy M. On Third Order Hyperbolic Jacobsthal Numbers, Konuralp Journal of
Mathematics. 2022;10(1):118-126.
Kuloglu B, O zkan E, Shannon AG. The Narayana Sequence in Finite Groups, Fibonacci Quarterly.
;60(5):212{221.
Soykan Y, Tasdemir E, Gocen M. Binomial Transform of the Generalized Third-Order Jacobsthal Sequence,
Asian-European Journal of Mathematics. 2022;15(12).
Available: https://doi.org/10.1142/S1793557122502242.
Soykan Y, Tasdemir E, Okumus, _I. A Study on Binomial Transform of the Generalized Padovan Sequence,
Journal of Science and Arts. 2022;22(1), 63-90.
Available: https://doi.org/10.46939/J.Sci.Arts-22.1-a06
Soykan Y. On Dual Hyperbolic Generalized Fibonacci Numbers. Indian J Pure Appl Math; 2021.
Available: https://doi.org/10.1007/s13226-021-00128-2
Soykan Y, Ta sdemir E, Okumus _I, Gocen M. Gaussian Generalized Tribonacci Numbers. Journal of
Progressive Research in Mathematics(JPRM). 2018;14 (2):2373-2387.
Soykan Y, Okumus _I, Tasdemir E. On Generalized Tribonacci Sedenions, Sarajevo Journal of Mathematics,
;16(1):103-122.
ISSN 2233-1964.
DOI: 10.5644/SJM.16.01.08
Soykan Y. Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, Communications in Mathematics
and Applications. 2020;11(2):281-295.
DOI: 10.26713/cma.v11i2.1102
Soykan Y. Explicit Euclidean Norm, Eigenvalues, Spectral Norm and Determinant of Circulant Matrix with
the Generalized Tribonacci Numbers, Earthline Journal of Mathematical Sciences. 2021;6(1):131-151.
Available: https://doi.org/10.34198/ejms.6121.131151
Soykan Y, Tetranacci and Tetranacci-Lucas Quaternions, Asian Research Journal of Mathematics.
;15(1): 1-24; Article no.ARJOM.50749.
Soykan Y. Gaussian Generalized Tetranacci Numbers, Journal of Advances in Mathematics and Computer
Science. 2019;31(3):1-21, Article no.JAMCS.48063.
Soykan; Asian Res. J. Math., vol. 19, no. 1, pp. 1-22, 2023; Article no.ARJOM.95530
Soykan Y. Matrix Sequences of Tetranacci and Tetranacci-Lucas Numbers, Int. J. Adv. Appl. Math. and
Mech. 2019;7(2):57-69. (ISSN: 2347-2529).
Soykan Y. On Binomial Transform of the Generalized Tetranacci Sequence, International Journal of
Advances in Applied Mathematics and Mechanics. 2021;9(2):8-27.
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