On doubled and quadrupled Fibonacci type sequences
Abstract
In this paper we study a family of doubled and quadrupled Fibonacci type sequences obtained by distance generalization of Fibonacci sequence. In particular we obtain doubled Fibonacci sequence, doubled and quadrupled Padovan sequence and quadrupled Narayana’s sequence. We give a binomial direct formula for these sequences using graph methods, and also we derive a number of identities. Moreover, we study matrix generators of these sequences and determine connections with the Pascal’s triangle.
Keywords
Fibonacci numbers; Padovan’s numbers; generalized Fibonacci numbers; Narayana’s numbers; generating function; Pascal’s triangle
References
J. Atkins and R. Geist, Fibonacci numbers and computer algorithms, College Math. J. 18 (1987), no. 4, 328–336.
U. Bednarz, A. Włoch, and M. Wołowiec-Musiał, Distance Fibonacci numbers, their interpretations and matrix generators, Comm. Math. 53 (2013), no 1, 35–46.
U. Bednarz, I. Włoch, and M. Wołowiec-Musiał, Total graph interpretation of the numbers of the Fibonacci type, J. Appl. Math. (2015), Art. ID 837917, 7 pp.
D. Bród, K. Piejko, and I. Włoch, Distance Fibonacci numbers, distance Lucas numbers and their applications, Ars Combin. 112 (2013), 397–409.
D. Bród and A. Włoch, (2,k)-distance Fibonacci polynomials, Symmetry 13 (2021), no. 2, Paper No. 303, 10 pp.
R. Diestel, Graph Theory, Springer-Verlag, Heidelberg-New York, 2005.
M.C. Er, Sums of Fibonacci numbers by matrix methods, Fibonacci Quart. 22 (1984), no. 3, 204–207.
S. Falcón, Binomial transform of the generalized k-Fibonacci numbers, Comm. Math. Appl. 10 (2019), no. 3, 643–651.
S. Falcón and Á. Plaza, The k-Fibonacci sequence and the Pascal 2-triangle, Chaos Solitons Fractals 33 (2007), no. 1, 38–49.
A.J. Feingold, A hyperbolic GCM Lie algebra and the Fibonacci numbers, Proc. Amer. Math. Soc. 80 (1980), no. 3, 379–385.
M.L. Fredman and R.E. Tarjan, Fibonacci heaps and their uses in improved network optimization algorithms, J. Assoc. Comput. Mach. 34 (1987), no. 3, 596–615.
A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly 68 (1961), 455–459.
D.V. Jaiswal, On a generalized Fibonacci sequence, Labdev–J. Sci. Tech. Part A 7 (1969), 67–71.
E. Kiliç, The generalized order-k Fibonacci-Pell sequence by matrix methods, J. Comput. Appl. Math. 209 (2007), no. 2, 133–145.
T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, New York, 2001.
M. Kwaśnik and I. Włoch, The total number of generalized stable sets and kernels in graphs, Ars Combin. 55 (2000), 139–146.
G.-Y. Lee, S.-G. Lee, J.-S. Kim, and H.-K. Shin, The Binet formula and representations of k-generalized Fibonacci numbers, Fibonacci Quart. 39 (2001), no. 2, 158–164.
N.Y. Li and T. Mansour, An identity involving Narayana numbers, European J. Combin. 29 (2008), no. 3, 672–675.
E. Özkan, B. Kuloğlu, and J.F. Peters, k-Narayana sequence self-similarity. Flip graph views of k-Narayana self-similarity, Chaos Solitons Fractals 153 (2021), Paper No. 111473, 11 pp.
E. Özkan, N. Şeyma Yilmaz, and A. Włoch, On F_3(k,n)-numbers of the Fibonacci type, Bol. Soc. Mat. Mex. (3) 27 (2021), no. 3, Paper No. 77, 18 pp.
G. Sburlati, Generalized Fibonacci sequences and linear congruences, Fibonacci Quart. 40 (2002), no. 5, 446–452.
M. Schork, Generalized Heisenberg algebras and k-generalized Fibonacci numbers, J. Phys. A 40 (2007), no. 15, 4207–4214.
N.J.A. Sloane, The On-Line Encyclopedia of Integer Sequences. Available at https://oeis.org/book.html
J. de Souza, E.M.F. Curado, and M.A. Rego-Monteiro, Generalized Heisenberg algebras and Fibonacci series, J. Phys. A 39 (2006), no. 33, 10415–10425.
Y. Soykan, Generalized Fibonacci numbers: sum formulas, J. Adv. Math. Comp. Sci. 35 (2020), no. 1, 89–104.
I. Stojmenovic, Recursive algorithms in computer science courses: Fibonacci numbers and binomial coefficients, IEEE Trans. Educ. 43 (2000), no. 3, 273–276.
S. Wagner and I. Gutman, Maxima and minima of the Hosoya index and the Merrifield-Simmons index: a survey of results and techniques, Acta Appl. Math. 112 (2010), no. 3, 323–346.
A. Włoch, Some identities for the generalized Fibonacci numbers and the generalized Lucas numbers, Appl. Math. Comput. 219 (2013), no. 10, 5564–5568.
I. Włoch, U. Bednarz, D. Bród, A. Włoch, and M. Wołowiec-Musiał, On a new type of distance Fibonacci numbers, Discrete Appl. Math. 161 (2013), no. 16-17, 2695–2701.
I. Włoch and A. Włoch, On some multinomial sums related to the Fibonacci type numbers, Tatra Mt. Math. Publ. 77 (2020), 99–108.
Graduate School of Natural and Applied Sciences, Erzincan Binali Yıldırım University Turkey
Wydział Matematyki i Fizyki Stosowanej, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
https://orcid.org/0000-0003-1789-6280
Department of Mathematics, Faculty of Arts and Sciences, Erzincan Binali Yıldırım University Turkey
Wydział Elektrotechniki i Informatyki, Politechnika Rzeszowska im. Ignacego Łukasiewicza Poland
This work is licensed under a Creative Commons Attribution 4.0 International License.
The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.
- License
This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license. - Author’s Warranties
The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s. - User Rights
Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor. - Co-Authorship
If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.
