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The reciprocal sums of the Fibonacci 3subsequences
Advances in Difference Equations volume 2016, Article number: 27 (2016)
Abstract
A Fibonacci 3subsequence 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 3subsequences and obtain several interesting families of identities involving the Fibonacci numbers.
1 Introduction
The Fibonacci sequence is defined by the linear recurrence relation
where \(F_{n}\) is called the nth Fibonacci number with \(F_{0}=0\) and \(F_{1}=1\). There exists a simple and nonobvious formula for the Fibonacci numbers,
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
For all \(n\geq2\),
Theorem 1.2
For all \(n\geq1\),
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 \(n\geq4\),
$$ \Biggl\lfloor \Biggl(\sum_{k=n}^{2n} \frac{1}{F_{k}} \Biggr)^{1} \Biggr\rfloor =F_{n2}. $$(1.3) 
(ii)
If \(m\geq3\) and \(n\geq2\), then
$$ \Biggl\lfloor \Biggl(\sum_{k=n}^{mn} \frac{1}{F_{k}} \Biggr)^{1} \Biggr\rfloor = \left \{ \textstyle\begin{array}{l@{\quad}l} F_{n2} &\textit{if }n \textit{ is even}, \\ F_{n2}1 &\textit{if }n \textit{ is odd}. \end{array}\displaystyle \right . $$(1.4)
Theorem 1.4
For all \(m\geq2\) and \(n\geq1\), we have
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
We have
Theorem 1.6
For all \(n\geq1\) and \(m\geq2\), we have
Theorem 1.7
If \(n\geq1\) and \(m\geq2\), then
Theorem 1.8
For all \(n\geq1\) and \(m\geq2\), we have
In this article, applying elementary methods, we investigate the reciprocal sums of the Fibonacci 3subsequences, 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 wellknown results on Fibonacci numbers, which will be used throughout the article. The detailed proofs can be found in [5].
Lemma 2.1
For any positive integers a and b, we have
As a consequence of (2.1), we have the following conclusion.
Corollary 2.2
If \(n\geq1\), then
The following interesting identity concerning the Fibonacci numbers plays a central role in the proofs of our main results.
Lemma 2.3
Assume that a, b, c are given nonnegative integers with \(a\geq b\). For \(n\geq a+c\), we have
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 \(n+1\).
First, it is easy to check that
where the last equality follows from (2.1).
Now we have
Similarly, we get \(F_{n+1+b}F_{n+1bc}=F_{2nc}+F_{n1+b}F_{n1bc}\).
Therefore, by the induction hypothesis, we arrive at
which completes the induction on n. □
Before introducing our main results, we first establish an inequality.
Lemma 2.4
If \(n\geq8\), then
Proof
It follows from (2.3) that
Since \(n\geq8\), we have
Thus, \(F_{n2}F_{n}F_{n+1}>2F_{n+2}F_{n+1}>2F_{2n+1}\). □
Theorem 2.5
For all \(n\geq2\),
Proof
Setting \(n=3k\), \(a=1\), \(b=0\), and \(c=2\) in (2.4), we obtain
It is straightforward to check that, for \(k\geq1\),
where the last equality follows from (2.7).
Therefore, we get that
If n is even, it is clear that
thus,
Now we consider the case where n is odd. It follows from (2.5) and the condition \(n\geq3\) that
from which we derive
Hence, we always have
irrespective of the parity of n.
If we let \(n=3k\), \(a=1\), \(b=0\), and \(c=1\) in (2.4), then
By elementary manipulations and (2.10) we deduce that, for \(k\geq1\),
which implies that
It follows from (2.2) that
from which we get that
Combining (2.9) and (2.11), we have
which yields the desired identity. □
We now study a generalization of Theorem 2.5 and start with an inequality.
Lemma 2.6
If \(m\geq3\) and \(n\geq3\), then
Proof
Letting \(a=b=n1\) in (2.1), we obtain \(F_{n}F_{n}< F_{2n1}\). If we set \(a=2n1\) and \(b=n+1\) in (2.1), then \(F_{2n1}F_{n+1}< F_{3n+1}\). Therefore,
from which we derive that
which completes the proof. □
Theorem 2.7
If \(m\geq3\) and \(n\geq1\), then
Proof
Applying the same argument as in the proof of Theorem 2.5, it is easy to see that if n is even, then
and thus the statement is true when n is even. We now concentrate ourselves on the case where n is odd.
Invoking (2.10), a direct calculation shows that, for \(k\geq 1\),
Hence, we arrive at
Employing (2.8) and the fact that n is odd, we conclude that
where the last inequality follows from Lemma 2.6.
It follows from (2.14) and (2.15) that if \(m\geq3\) and n is odd, then
from which the desired identity follows immediately. □
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
For all \(n\geq1\),
Theorem 2.9
If \(m\geq3\) and \(n\geq1\), then
Theorem 2.10
For all \(n\geq1\),
Theorem 2.11
If \(m\geq3\) and \(n\geq1\), then
3 Main results II: the reciprocal square sums
In the rest of the article, we study the reciprocal square sums of the Fibonacci 3subsequences.
Lemma 3.1
Let a and b be two given integers with \(a\geq b\geq0\). For all \(n\geq a\), we have
Proof
Employing (2.4), we derive that
The proof is complete. □
Lemma 3.2
If \(n\geq1\), we have
Proof
First, it is easy to see that
Similarly, we have
It follows from (2.1) that \(F_{6n+1}>F_{3n}^{2}+1\) and \(F_{6n+2}>F_{3n+1}F_{3n+2}+1\). Therefore,
The desired result follows immediately from (3.3), (3.4), and (3.5). □
Lemma 3.3
For \(n\geq1\), we have
Proof
It is easy to see that \(8F_{3n}^{2}16>F_{3n}^{2}F_{3n3}^{2}\) and \(8F_{3n+3}^{2}+16< F_{3n+6}^{2}F_{3n+3}^{2}\), so
and
Hence,
which completes the proof. □
We now present the first reciprocal square sum of the Fibonacci 3subsequence.
Theorem 3.4
If \(n\geq2\) and \(m\geq2\), then
Proof
We first consider the case where n is even. Applying (3.1), by a direct calculation we get that, for \(k\geq1\),
from which we get
Since n is even, it is easy to see that
which implies that
A similar manipulation yields that, for \(k\geq1\),
where the last inequality follows from the easily checked fact
Now we have
where the last inequality follows from (3.2).
Combining (3.8) and (3.9), we get that
which means that the statement is true when n is even.
We now consider the case where n is odd. It is not hard to derive that
where the last inequality follows from the fact that \(F_{3k+3}=3F_{3k}+2F_{3k1}\).
Therefore, we arrive at
Invoking (3.7) and Lemma 3.3, we derive that if n is odd, then
where the last inequality follows from the fact that \(F_{6n+2}>F_{3n}F_{3n+3}\) and \(2F_{6n+1}>F_{6n+2}\).
It follows from (3.10) and (3.11) that
which yields the desired result. □
For the subsequences \(\{F_{3k+1}\}\) and \(\{F_{3k+2}\}\), we have similar results.
Theorem 3.5
If \(n\geq1\) and \(m\geq2\), then
Theorem 3.6
If \(n\geq1\) and \(m\geq2\), then
Remark
The proof of Theorem 3.5 is similar to that of Theorem 3.4 and is omitted here. Since the telescoping technique for the proof of Theorem 3.6 is very different from that for Theorem 3.4, we give a detailed proof of Theorem 3.6 in the next section.
4 Proof of Theorem 3.6
We first present a preliminary result, which plays a central role in the later proof.
Lemma 4.1
For all \(n\geq1\), we have
Proof
The second inequality is obvious, so we concentrate ourselves on the first one. It is easy to see that
Employing (2.1), we get \(F_{6n+4}=F_{3n+2}(F_{3n+1}+F_{3n+3})\); therefore,
It follows from \(2F_{3n+1}>F_{3n+2}=F_{3n+5}2F_{3n+3}\) that \(2(F_{3n+1}+F_{3n+3})>F_{3n+5}\); thus,
Combining (4.1), (4.2), and (4.3) yields the desired result. □
Proof of Theorem 3.6
We first consider the case where n is even. Employing (3.1) again, by a direct calculation we deduce that, for \(k\geq1\),
from which we obtain
Since n is even, it is easy to see that
which implies that
Similarly, we can derive that, for \(k\geq1\),
We now have
where the last inequality follows from Lemma 4.1.
Combining (4.5) and (4.6) yields that
which shows that the statement is true when n is even.
Next we turn to consider the case where n is odd. It is not hard to derive that
where the last inequality follows from the fact that \(F_{3k+5}=3F_{3k+2}+2F_{3k+1}\).
Therefore, we obtain
Applying (4.4) and the fact that n is odd, we derive that
where the last inequality follows from Lemma 4.1.
It follows from (4.7) and (4.8) that
which yields the desired result. □
5 Conclusions
In this paper, we investigate in two ways the reciprocal sums of the Fibonacci 3subsequences, where a Fibonacci 3subsequence 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.
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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).
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All authors contributed equally to deriving all the results of this article and read and approved the final manuscript.
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Wang, A.Y., Zhang, F. The reciprocal sums of the Fibonacci 3subsequences. Adv Differ Equ 2016, 27 (2016). https://doi.org/10.1186/s1366201607612
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DOI: https://doi.org/10.1186/s1366201607612