A generalization of Lucas sequence and associated identities
DOI:
https://doi.org/10.5269/bspm.53068Abstract
In this paper, we attempt to generalize Lucas sequence by generating certain number of sequences whose terms are obtained by adding the last two generated terms of the preceding sequence. Lucas sequence is obtained as a particular case of generating only one sequence. Moreover we prove some of the results which can be seen as generalized form of the results which hold for Lucas sequence. We obtain Cassini-like identity for these generalized Lucas sequences.
References
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2. D. M. Burton, Elementary Number Theory, Seventh edition, McGraw-Hill Education, (2015).
3. Y. Choo, On the generalizations of Fibonacci identities, Results in Mathematics, 71, 347-356, (2017). https://doi.org/10.1007/s00025-015-0515-6
4. Gregory P. B. Dresden and Zhaohui Du, A simplified Binet formula for k-generalized Fibonacci numbers, Journal of Integer Sequences, Vol. 17, 1-9, (2014).
5. M. C. Er, Sums of Fibonacci numbers by matrix methods, The Fibonacci Quarterly, 22(3), 204-207, (1984).
6. G. H. Hardy and E. M. Wright, An Introduction To The Theory Of Numbers, Sixth edition, Oxford University Press, (2008).
7. F. B. Hildebrand, Introduction To Numerical Analysis, Second edition, Dover Publications, (2016).
8. Y. Kwon, A note on the modified k-Fibonacci-like sequence, Commun. Korean Math. Soc., 31, No. 1, 1-16, (2016). https://doi.org/10.4134/CKMS.2016.31.1.001
9. E. P. Miles, Jr., Generalized Fibonacci numbers and associated matrices, The American Mathematical Monthly, Vol. 67, No. 8, 745-752, (Oct., 1960). https://doi.org/10.2307/2308649
10. N. K. Paul and H. K. Saikia, A generalization of Fibonacci sequence, Proyecciones (Antofagasta, On line), vol. 39, no. 6, pp. 1393-1405, (Dec. 2020). https://doi.org/10.22199/issn.0717-6279-2020-06-0085
11. N. K. Paul and H. K. Saikia, Some generalized results related to Fibonacci sequence, Proyecciones (Antofagasta, On line), (In press).
12. A. P. Stakhov, Fibonacci matrices, a generalization of the "Cassini formula", and a new coding theory, Chaos, Solitons and Fractals, 30, 56-66, (2006). https://doi.org/10.1016/j.chaos.2005.12.054
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2022-12-23
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