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Accessible Unlicensed Requires Authentication Published by De Gruyter June 7, 2017

A monotonicity property for generalized Fibonacci sequences

Toufik Mansour and Mark Shattuck
From the journal Mathematica Slovaca

Abstract

Given k ≥ 2, let an be the sequence defined by the recurrence an = α1an–1 + … + αkank for nk, with initial values a0 = a1 = … = ak–2 = 0 and ak–1 = 1. We show under a couple of assumptions concerning the constants αi that the ratio annan1n1 is strictly decreasing for all nN, for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the αi are unity or when all of the αi are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.


(Communicated by Federico Pellarin)


References

[1] Benjamin, A. T.—Quinn, J. J.: Proofs that Really Count: The Art of Combinatorial Proof, Mathematical Association of America, 2003.Search in Google Scholar

[2] Brenti, F.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry: an update, Contemp. Math. 178 (1994), 71–89.Search in Google Scholar

[3] Chen, W. Y. C.—Guo, J. J. F.—Wang, L. X. W.: Zeta functions and the log-behavior of combinatorial sequences, Proc. Edinb. Math. Soc. 58 (2015), 637–651.Search in Google Scholar

[4] Hou, Q.-H.—Sun, Z.-W.—Wen, H.: On monotonicity of some combinatorial sequences, Publ. Math. Debrecen 85 (2014), 285–295.Search in Google Scholar

[5] Knuth, D. E.: The Art of Computer Programming: Sorting and Searching, Vol. 3, Addison-Wesley, 1973.Search in Google Scholar

[6] Liu, L. L.—Wang, Y.: On the log-convexity of combinatorial sequences, Adv. Appl. Math. 39 (2007), 453–476.Search in Google Scholar

[7] Mansour, T.—Shattuck, M.: Polynomials whose coefficients are k-Fibonacci numbers, Ann. Math. Inform. 40 (2012), 57–76.Search in Google Scholar

[8] Munarini, E.: A combinatorial interpretation of the generalized Fibonacci numbers, Adv. Appl. Math. 19 (1998), 306–318.Search in Google Scholar

[9] Sloane, N. J. A.: The On-Line Enyclopedia of Integer Sequences, available at , 2010.Search in Google Scholar

[10] Stanley, R. P.: Log-concave and unimodal sequences in algebra, combinatorics, and geometry, Ann. New York Acad. Sci. 576 (1989), 500–534.Search in Google Scholar

[11] Sun, Z.-W.: On a sequence involving sums of primes, Bull. Aust. Math. Soc. 88 (2013), 197–205.Search in Google Scholar

[12] Sun, Z.-W.: Conjectures involving arithmetical sequences. In: Number Theory: Arithmetic in Shangri-La, Proceedings of the 6th China-Japan Seminar (Shanghai, 2011), World Scientific, 2013, pp. 244–258.Search in Google Scholar

[13] Wang, Y.—Zhu, B. X.: Proofs of some conjectures on monotonicity of number theoretic and combinatorial sequences, Sci. China Math. 57 (2014), 2429–2435.Search in Google Scholar

Received: 2014-10-19
Accepted: 2015-10-11
Published Online: 2017-6-7
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences