Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

A Nonlinear Extension of Fibonacci Sequence

M. Tamba*, Y.S. Valaulikar

Department of Mathematics, Goa University, Taleigaon Plateau, Goa, India

Abstract: A new extension of Fibonacci sequence which yields a nonlinear second order recurrence relation is defined. Some identities and congruence properties for the new sequence are obtained.

Keywords: fibonacci sequence, nonlinear recurrence relation, congruence properties

Citation: M. Tamba, Y.S. Valaulikar, A Nonlinear Extension of Fibonacci Sequence. Turkish Journal of Analysis and Number Theory, 4 (4). 109-112. January 2016. doi:10.12691/tjant-4-4-4

1. Introduction

T he well-known Fibonacci sequence F(n) is defined by

[picture] | (1.1)

is called the nth Fibonacci number. The Fibonacci sequence has been extended in many ways and it has many interesting properties and applications in different fields ([5,6]). In recent years, Fibonacci numbers are also seen in many combinatorial problems ([1,3]).

In this note, we present yet another extension of Fibonacci sequence and call it Fibosenne sequence defined by the relation

[picture] | (1.2)

where is the nth Fibonacci number. We call the n th Fibosenne number in view of its form like Mersenne number [4]. It is clear that . We shall establish various relations for in line with those of . We shall also study some congruence properties of .

2. Fibosenne Sequence {P(n)}

For , let denote the Fibosenne number. Then the following results follow immediately from definition (1.2) and from results on Fibonacci number ([2,7]).

Proposition 2.1

1. For ,

where are the roots of the equation

2. For , satisfies the nonlinear second order recurrence relation,

[picture] | (2.1)

with initial conditions .

3. For ,

Proof.

1. Follows from definition and Binet’s formula for

2. Follows from definition.

3. Follows from definition and

The following properties can also be deduced easily. These properties are useful to find using the earlier Fibosenne numbers.

Proposition 2.2 We have

1.

2.

3. For

4. For

5.

Proof

1. Using definition of and identity 18 for given in ([2]), we get,

Hence the result follows.

2. Using (1),

By recurrence relation (2.1), we see that

Combining 1. and 2. obtained in this proof gives desired result.

3. We have for that

(See [2] and [7]). Thus we arrive at the desired result.

4. Using the following identity and definition of result follows.

The following results are concerning divisibility of

Proposition 2.3

1. For .

2. If .

3. If .

4. If is a prime such that

Proof.

1. . Suppose , Then there is a prime such that divides both and . Hence by equation (2.1), divides . Inductively, divides , giving contradiction.

Proofs of (2)- (4) follows from the fact that, .

3. Some Congruence Properties of Fibosenne Sequence {P(n)}

In this section, we present some congruence properties of Fibosenne sequence. The following table gives Fibosenne numbers for .

A look at the Table 1 reveals that the last digit of P(n) follow the pattern 0113715, 113715, ... . Given , we have and . For , the congruence properties of P(n) can be found by using the recurrence relation (2.1). For , we show the cycles of in Table 2.

Table 1. P(n), 0≤n≤12

Table 2. Cycles of P(n) (mod m), b(m), t(m), 2≤m≤10

Here we observe that P(n) modulo m show some pattern. After initial terms which we call “Base” there is periodic repetition of terms. We call it “Tail”. This suggests that there is a pattern for congruence residues, which is different from that of Fibonacci sequence. For all sequences, which show such a pattern for congruence modulo m, we define the Base length as the number of terms in the base and denote it by b(m), the base length of the sequence modulo m. Similarly, we define the Tail Period as the minimum number of terms repeating in the tail of the sequence modulo m and denote it by t(m). For example when , the sequence is 0, 1, 1,3,3,.... Here 0,1,1 is the base and 3,3,3,... the tail. There are three terms in the base and hence b(4)= 3 . In the tail number 3 repeats and so t(4)= 1. In the second column of Table 2, base is shown in brackets. If there are no initial terms forming base, then b(m) = 0 and the bracket is omitted. It is noteworthy that for m = 11, b(11) = 46 and t(11) =12.

From Table 2, for m = 10, the following result follows immediately.

Proposition 3.1. For

The following result also from Table 2.

Proposition 3.2

1. For and

2. For and

For let be the largest such that For example, etc.

Proposition 3.3 For , for . In this case and

Proof. For .

[picture] | (3.1)

Given considering the Fibonacci numbers and the smallest residues of the terms modulo , it was observed ([9], Chap.VII) that the sequence repeats after term as are given in ([9]) which we tabulate here:

Table 3. Values of π(m) for 2≤m≤30

is called pisano period of modulo . From the definition of , it is clear that

Lemma 3.4 For and ,

Using this lemma we prove the following:

Proposition 3.5 For and , we have

and for some then

Proof. We have

[picture] | (3.2)

Notice that in Proposition 3.5, suggest that is odd. The case when is power of 2 is dealt in Proposition 3.3. When is even, but not power of 2, we have the following:

Proposition 3.6 For and , if

where and, for some , then for where is independent of u. Here .

Proof. We see for that

[picture] | (3.3)

where is the remainder when is divided by .

We have the following corollaries.

Corollary 3.7. For we have

Proof. Follows from Proposition 3.5 as and taking .

Corollary 3.8. For ,

Proof. As so taking Proposition 3.5,

When .

When .

Hence, by (1.2), for .

Similarly the congruence relations for can be obtained recursively.

Corollary 3.9. For

Proof. As taking and using Proposition 3.5 and proceeding as in the proof of Corollary 3.8 the result follows.

We have the following results on the tail period.

Proposition 3.10

1. If is an odd integer,then and , where is Euler’s - function. In particular, if is prime, then

2. For

3. Let be an integer such that then

Proof

1.

Thus the pattern in repeats after terms. So must be a multiple of Also since the cycle repeats right from the beginning,

2. For

3.

Using result (3) in Proposition 3.10, we have the following properties.

Proposition 3.11

1. For if then .

2. For if then .

3. If is the largest integer such that and for some prime , then , for all

4. If is a prime and , then .

5. If is a prime of the form , then

6. If prime is of the form and , then

Proof

1. For if Hence by Proposition 3.10 (3), . Result now follows by ([8], Prop 3.5).

2. Follows as in (1) above from Proposition 3.10 (3) and ([8], Prop 3.7).

3. Follows as in (1) above from Proposition 3.10 (3) and ([8], Prop 3.8).

4. If is a prime and , then

Then by Proposition 3.10 (3), . Hence by ([8], Prop 3.11).

5. Follows as in (4) above from Proposition 3.10 (3) and ([8], Prop 3.12).

6. Follows as in (4) above from Proposition 3.10 (3) and ([8], Prop 3.13).

4. Conclusion

The new sequence defined using nonlinear second order recurrence relation has most of the identities satisfied by Fibonacci sequence. However the congruence properties of this sequence are different from those of Fibonacci sequence.

References

[1]   | Z.Akyuz, S. Halici, On Some Combinatorial Identities involving the terms of generalized Fibonacci and Lucas sequences, Hacettepe Journal of Mathematics and Statistics, Volume 42 (4) (2013), 431-435.
[2]   | A.T. Benjmin, J.J. Quinn, Proofs that really count: The Art of Combinatorial Proof, Mathematical Association of America, Washington, D.C., 2003.
[3]   | A.T. Benjmin, J.J. Quinn, The Fibonacci Numbers Exposed More Discretely, Mathematics Magazine 76:3 (2003), 182-192.
[4]   | D. Burton, Elementary Number Theory, 6th edition, Tata McGraw-Hill, 2006.
[5]   | M. Edson, O. Yayenie, A new generalization of Fibonacci sequence and extended Binet's formula, Integers, Volume 9, Issue 6, Pages 639-654, ISSN (Print) 1867-0652.
[6]   | J. Kappraff, G.W. Adamson, Generalized Binet Formulas, Lucas polynomials and Cyclic constants, Forma,19,(2004) 355-366.
[7]   | M. Renault, The Fibonacci sequence under various moduli, Masters Thesis, 1996.
[8]   | S. Vajda, Fibonacci and Lucas numbers and the Golden section: Theory and Applications, Dover Publications, 2008.