The reciprocal sums of the Fibonacci 3-subsequences
- Andrew YZ Wang^{1}Email author and
- Fan Zhang^{1}
https://doi.org/10.1186/s13662-016-0761-2
© Wang and Zhang 2016
Received: 7 December 2015
Accepted: 18 January 2016
Published: 26 January 2016
Abstract
A Fibonacci 3-subsequence is a subsequence of the type \(F_{n},F_{n+3},F_{n+6},\ldots\) , where \(F_{k}\) denotes the kth Fibonacci number. In this article, we investigate the reciprocal sums of the Fibonacci 3-subsequences and obtain several interesting families of identities involving the Fibonacci numbers.
Keywords
MSC
1 Introduction
The Fibonacci sequence plays an important role in the theory and applications of mathematics, and its various properties have been investigated by many authors: see [1–5].
In recent years, there has been an increasing interest in studying the reciprocal sums of the Fibonacci numbers. For example, Elsner, Shimomura, and Shiokawa [6–9] investigated the algebraic relations for reciprocal sums of the Fibonacci numbers. Ohtsuka and Nakamura [10] studied the partial infinite sums of the reciprocal Fibonacci numbers. They established the following results, where \(\lfloor\cdot\rfloor\) denotes the floor function.
Theorem 1.1
Theorem 1.2
Recently, Wang and Wen [11] considered the partial finite sum of the reciprocal Fibonacci numbers and strengthened Theorem 1.1 and Theorem 1.2 to the finite sum case.
Theorem 1.3
- (i)For all \(n\geq4\),$$ \Biggl\lfloor \Biggl(\sum_{k=n}^{2n} \frac{1}{F_{k}} \Biggr)^{-1} \Biggr\rfloor =F_{n-2}. $$(1.3)
- (ii)If \(m\geq3\) and \(n\geq2\), then$$ \Biggl\lfloor \Biggl(\sum_{k=n}^{mn} \frac{1}{F_{k}} \Biggr)^{-1} \Biggr\rfloor = \left \{ \textstyle\begin{array}{l@{\quad}l} F_{n-2} &\textit{if }n \textit{ is even}, \\ F_{n-2}-1 &\textit{if }n \textit{ is odd}. \end{array}\displaystyle \right . $$(1.4)
Theorem 1.4
Furthermore, the present authors [12] studied the reciprocal sums of even and odd terms in the Fibonacci sequence and obtained the following main results.
Theorem 1.5
Theorem 1.6
Theorem 1.7
Theorem 1.8
In this article, applying elementary methods, we investigate the reciprocal sums of the Fibonacci 3-subsequences, by which we mean the subsequences of the type \(F_{n},F_{n+3}, F_{n+6},\ldots\) and obtain several interesting families of identities involving the Fibonacci numbers.
2 Main results I: the reciprocal sums
We first present several well-known results on Fibonacci numbers, which will be used throughout the article. The detailed proofs can be found in [5].
Lemma 2.1
As a consequence of (2.1), we have the following conclusion.
Corollary 2.2
The following interesting identity concerning the Fibonacci numbers plays a central role in the proofs of our main results.
Lemma 2.3
Proof
We proceed by induction on n. It is clearly true for \(n=a+c\). Assuming that the result holds for any integer \(n\geq a+c\), we show that the same is true for \(n+1\).
Before introducing our main results, we first establish an inequality.
Lemma 2.4
Proof
Theorem 2.5
Proof
We now study a generalization of Theorem 2.5 and start with an inequality.
Lemma 2.6
Proof
Theorem 2.7
Proof
Applying a similar analysis to the subsequences \(\{F_{3k+1}\}\) and \(\{ F_{3k+2}\}\), we obtain the following results, whose proofs are omitted here.
Theorem 2.8
Theorem 2.9
Theorem 2.10
Theorem 2.11
3 Main results II: the reciprocal square sums
In the rest of the article, we study the reciprocal square sums of the Fibonacci 3-subsequences.
Lemma 3.1
Proof
Lemma 3.2
Proof
Lemma 3.3
Proof
We now present the first reciprocal square sum of the Fibonacci 3-subsequence.
Theorem 3.4
Proof
For the subsequences \(\{F_{3k+1}\}\) and \(\{F_{3k+2}\}\), we have similar results.
Theorem 3.5
Theorem 3.6
4 Proof of Theorem 3.6
We first present a preliminary result, which plays a central role in the later proof.
Lemma 4.1
Proof
Combining (4.1), (4.2), and (4.3) yields the desired result. □
Proof of Theorem 3.6
5 Conclusions
In this paper, we investigate in two ways the reciprocal sums of the Fibonacci 3-subsequences, where a Fibonacci 3-subsequence is a subsequence of the type \(F_{n},F_{n+3}, F_{n+6},\ldots\) . One is focused on the ordinary sums, and the other is concerned with the square sums. By evaluating the integer parts of the reciprocals of these sums we get several interesting families of identities. The results are new and important to those with closely related research interests.
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for their helpful comments. This work was supported by the National Natural Science Foundation of China (No. 11401080).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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