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2046-1402

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10.12688/f1000research.157738.2

Research Article

Articles

More on the Fascinating Characterizations of Mulatu’s Numbers

[version 2; peer review: 4 approved with reservations, 2 not approved]

Derso

Derebew Nigussie

Conceptualization Formal Analysis Funding Acquisition Investigation Methodology Project Administration Resources Software Supervision Validation Visualization Writing – Original Draft Preparation Writing – Review & Editing https://orcid.org/0000-0003-2431-2802 1 Admasu

Ageze Abye

a ageze.ab19@gmail.com

No competing interests were disclosed.

17 4 2025

2024

1306

7 4 2025

2025

This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Background

The discoveries of Mulatu’s numbers, better known as Mulatu’s sequence, represent revolutionary contributions to the mathematical world. His best-known work is Mulatu’s sequence, in which each new number is the sum of the two preceding numbers. When various operations and manipulations are performed on the numbers in this sequence, remarkable and intricate patterns begin to emerge. This study aimed to identify novel characterizations of Mulatu’s numbers.

Methods

This study employed a multi-faceted approach to investigate characterizations of Mulatu’s numbers. Mathematical proof techniques such as principle of mathematical induction, proof by contradiction and direct proof were utilized to substantiate findings.

Results

In this study, we provided several characterizations of Mulatu’s numbers. We also investigated the properties and patterns of these numbers. Moreover, we have also shown that, similar to Fibonacci’s numbers, Mulatu’s numbers give the so-called golden ratio, which is most applicable in numerical optimization. Furthermore, we formulated a relation among Mulatu’s numbers, Fibonacci numbers, and Lucas numbers. Finally, we provided a generating function for the Mulatu numbers.

Conclusions

In this study, we uncovered novel characterizations of Mulatu’s numbers and introduced a generating function for them. We investigated relationship between Mulatu’s numbers and the golden ratio. The results discussed offer valuable insights and enhance our understanding of their properties. Furthermore, these findings play a vital role in the boarder context of mathematics and related areas, contributing significantly to the field.

Mulatu’s number; Mulatu’s sequence; γ-Mulatu summable; Mulatu’s series; Mulatu’s characteristics number; generating function.

Woldia University

This research was partially supported by Woldia University.

The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Revised Amendments from Version 1

We thank you and the reviewers for the valuable comments and suggestions on our manuscript titled “ More on the fascinating characterizations of Mulatu’s numbers ”. We have carefully revised the manuscript and addressed all feedbacks. Below is a brief summary of the changes made:

Introduction updated: We included sentences that more explain novelty of this work as suggested by reviewers. Some reference numbers are reshuffled.

Methods clarified: We included statements to clarify the GNU octave was used only for conceptualization purpose as requested by reviewers.

Discussion improved: We included the proof of lemma 1 as suggested by reviewers. Moreover, the proofs of theorem 1, 2 and 3 are updated as relatively shorter methods of proofs suggested by reviewers.

References: Some changes are made on references based on reviewers’ comments such as inclusion of appropriate sites.

Introduction

Mulatu numbers are a recently introduced sequence by Mulatu Lemma, a Professor of Mathematics at Savannah State University in Savannah, Georgia. ^{1,
2} These numbers have been introduced in different studies. ^{3,
4} Mulatu’s numbers defined as 4, 1, 5, 6, 11, 17, 28, 45, 73, …. Mathematically, such sequence can be written as follow. M n = { 4 if n = 0 1 if n = 1 M n − 1 + M n − 2 if n > 1

Mulatu Lemma’s work has sparked curiosity in many scholars, encouraging them to dive deeper into this fascinating field. He’s especially recognized for a special sequence of numbers that bears his name, known for its intriguing patterns. Mulatu’s contributions continue to inspire and engage anyone interested in the beauty of mathematics. ^{5,
6}

Mulatu’s sequence is a series of numbers where each new term is found by adding the two before it. It’s the latest example of a recursive sequence, meaning each term is built from the previous ones. Taking a closer look at Mulatu’s sequence opens the door to a wealth of fascinating patterns and mathematical properties. ⁵ Over the years, researchers have discovered many interesting trends within it. For instance, the Fibonacci sequence starts with 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and continues on, while the Lucas sequence begins with 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, and grows from there. ⁷ Both of these sequences showcase the elegance and intricacy of numbers.

Previous works also discussed many of the Fibonacci and Luca numbers, ^{2,
7–
9} but only some characterizations of Mulatu’s numbers. ^{2,
5,
6} Thus, in this paper, we focused on these less studied numbers, Mulatu’s numbers.

So as to fill this gap, we found novel characterizations of Mulatu numbers. Mulatu, Fibonacci and Lucas numbers are related using divisibility. We studied Mulatu numbers as a sequence, a number and a series as well. A generated function is found for the Mulatu sequence. Furthermore, Mulatu numbers and the set of natural numbers are related after phrases Mulatu summable, γ -Mulatu summable and Mulatu characteristic number are defined.

Regarding characterizations of Mulatu numbers, we have used the GNU Octave (version 4.0.0) ¹⁰ to conceptualize theorems by taking numerical examples before formal proofs are given.

We have produced new results on Mulatu’s numbers and on the interrelationships between Fibonacci’s and Lucas’s numbers. We are interested in the generic area of gradient-free optimization when derivative information for the function is unavailable or calculation of the derivatives is computationally difficult, and the Golden section search method is one such algorithm that uses the Golden ratio as an input. ¹¹ Hence, in addition, we related the Mulatu sequence with the so-called golden ratio, which is most applicable in numerical optimization, specifically to find an approximate optimizer with a small error.

Methods

In this study, we applied various mathematical proof techniques, including the principle of mathematical induction, proof by contradiction, and direct proof, to investigate multiple characterizations. To conceptualize and formulate conjectures, and conduct numerical examples prior to presenting formal proofs we utilized GNU Octave (version 4.0.0) software. ¹⁰ This combination of theoretical and computational approaches enabled a comprehensive exploration of Mulatu numbers.

Characterizations of Mulatu’s Numbers

Lemma 1.

Any two consecutive Mulatu numbers are relatively prime.

Proof.

Mathematically, the lemma is gcd ( M n , M n + 1 ) = 1 for any non-negative integer n , where gcd denotes greatest common divisor.

We use induction on n .

For n = 0 , gcd ( M 0 , M 1 ) = gcd ( 4 1 ) = 1 . Assume the lemma holds for n = k > 0 . That is gcd ( M k , M k + 1 ) = 1 , for all k > 0 .

Claim: gcd ( M k + 1 , M k + 2 ) = 1 , for all k ≥ 0 .

From the Euclidean algorithm, we have:

gcd ( M n , M n + 1 ) = gcd ( M n , M n + 1 mod M n ) , for all n ≥ 0 .

Now, M k + 2 mod M k + 1 = ( M k + M k + 1 ) mod M k + 1 = M k . This implies that

gcd ( M k + 1 , M k + 2 ) = gcd ( M k , M k + 1 ) = 1 , for all k ≥ 0 . Therefore, the lemma holds by the principle of mathematical induction.

Theorem 1.

Adding any ten consecutive Mulatu numbers together will always result in a number that is divisible by 11.

Proof.

Let q = ∑ n = i i + 9 M n for any number i . We then show that q is divisible by 11 .

Using the definition of Mulatu numbers, we have M i + j = F j − 1 M i + F j M i + 1 for j ≥ 2 in the set of natural numbers, where F n denotes the n th Fibonacci number. q = ∑ n = i i + 9 M n = M i + M i + 1 + M i + 2 + M i + 3 + … + M i + 9 = 55 M i + 88 M i + 1 .

This completes the proof.

Theorem 2.

Multiplying any Mulatu’s number by two and subtracting the next Mulatu’s number in the sequence for a number greater than or equal to two will result in the answer being Mulatu’s number, that is, 2 M n − M n + 1 = M n − 2 , n ≥ 2 .

Proof.

Claim: 2 M n − M n + 1 = M n − 2 , n ≥ 2 .

Now, M n + 1 + M n − 2 = M n + M n − 1 + M n − 2 = 2 M n .

Theorem 3.

When adding consecutive, even-positioned Mulatu numbers beginning with M 2 i , for i ≥ 1 , the result is a number that is one less than the Mulatu number succeeding the last Mulatu number in the sum. This provides a general formula for a simple way to find the sum of any finite even-positioned Mulatu number. ∑ i = 1 n M 2 i = M 2 n + 1 − 1 .

Proof.

By definition, M 2 i = M 2 i + 1 − M 2 i − 1 , for any natural number i.

This implies that ∑ i = 1 n M 2 i = ∑ i = 1 n ( M 2 i + 1 − M 2 i − 1 ) = ( M 3 − M 1 ) + ( M 5 − M 3 ) + ( M 7 − M 5 ) + ( M 9 − M 7 ) + … + ( M 2 n − 5 − M 2 n − 7 ) + ( M 2 n − 3 − M 2 n − 5 ) + ( M 2 n − 1 − M 2 n − 3 ) + ( M 2 n + 1 − M 2 n − 1 ) = M 2 n + 1 − M 1 = M 2 n + 1 − 1 .

So, we are done.

Theorem 4.

When adding consecutive, odd-positioned Mulatu’s numbers beginning with M 1 , only this time, the result is a number that is four less than the Mulatu number following the last even number in the sum. This provides a general formula for a simple way to find the sum of any finite odd-positioned Mulatu number. ∑ i = 1 n M 2 i − 1 = M 2 n − 4 .

Proof.

We use induction on n .

Theorem 5.

When any four consecutive numbers in the Mulatu sequence are considered, the difference between the squares of the two numbers in the middle is equal to the product of the two outer numbers. Mathematically, M n + 1 2 − M n 2 = M n − 1 M n + 2 , n ≥ 1 .

Corollary 1.

For n ≥ 1 , 2 M n + M n − 1 = M n + 2 .

Proof.

By using Theorem 5 above and the relation M n + 1 = M n + M n − 1 , the result follows.

Theorem 6.

Adding any number of consecutive Mulatu numbers will result in a number that is one less than the Mulatu number of two places beyond the last summand. This provides a general formula for a simple way to find the sum of any number of Mulatu numbers. ∑ i = 0 n M i = M n + 2 − 1 .

Theorem 7.

Let M n , F n and L n be the Mulatu’s, Fibonacci’s, and Lucas’s numbers for each n ≥ 0 . The number ( M n + M n + 3 ) ( F n + F n + 3 ) ( L n + L n + 3 ) is divisible by 8.

Proof.

We use induction on n .

For n = 0 , M 0 + M 3 = 10 , F 0 + F 3 = 2 and L 0 + L 3 = 6 . This implies that

( M 0 + M 3 ) ( F 0 + F 3 ) ( L 0 + L 3 ) = 120 is divisible by 8.

Assume that the statement holds for n = k . As M k + M k + 3 = 2 ( M k + M k + 1 ) , we have M k + 1 + M k + 4 = 2 ( M k + 2 M k + 1 ) .

Thus, 2 | ( M k + 1 + M k + 4 ) . Similarly, 2 | ( F k + 1 + F k + 4 ) and 2 | ( L k + 1 + L k + 4 ) .

Hence, the theorem holds by the principle of mathematical induction.

Theorem 8.

Half of the sum of any two even consecutive Mulatu numbers yields Mulatu’s number preceding the larger one. i.e., 1 2 ( M 3 n + M 3 ( n + 1 ) ) = M 3 ( n + 1 ) − 1 .

Proof.

The above equation represents the statement of the theorem as it is easy to show that every (3 n) ^th Mulatu number is even for any non-negative integer n.

We use induction on n .

For n = 0 , 1 2 ( M 0 + M 3 ) = 5 = M 2 . Let it be true for n = k . That is 1 2 ( M 3 k + M 3 ( k + 1 ) ) = M 3 ( k + 1 ) − 1 .

This assumption with the definition of Mulatu’s sequence leads:

1 2 ( M 3 n + M 3 ( n + 1 ) ) = M 3 ( n + 1 ) − 1 , for all n .

The following definitions are given to study Mulatu numbers in relation with natural numbers, and to define a characteristic number to it.

Definition 1.

a) Two natural numbers are said to be Mulatu summable if their sum is a Mulatu number.

b) A natural number k is said to be γ -Mulatu summable if a natural number γ ≥ 2 exists such that γk is a Mulatu number.

Example: 2 and 3 are Mulatu summable, but 3 and 5 are not Mulatu summable. Moreover, 1 is γ -Mulatu summable for all γ = M n ≠ 1 , where M n is the Mulatu number and 2 is γ -Mulatu summable for γ = 2 , 3 , 14 , . … Definition 2.

Let γ be the smallest natural number such that a natural number k is γ -Mulatu summable. Then, γ is said to be the Mulatu characteristic of k , denoted by m ∗ ( k ) .

Example: m ∗ ( 1 ) = 4 , m ∗ ( 2 ) = 2 , m ∗ ( 4 ) = 7 and so on. Lemma 2.

The Mulatu characteristics neither preserve nor reverse any inequality.

Proof.

Let k and n be natural numbers with k < n . Suppose that the Mulatu characteristic either preserve or reverse an inequality. That is, the Mulatu characteristics preserve or reverse an inequality. Thus, it must preserve or reverse the inequality k < n . Thus, m ∗ ( k ) < m ∗ ( n ) or m ∗ ( k ) > m ∗ ( n ) for each k < n .

Neither is true because, for instance, 2 < 3 but m ∗ (2) = 2 = m ∗ (3). For the remaining inequalities, we can see counter examples 1 < 2 but m ∗ ( 1 ) = 4 > 1 = m ∗ ( 2 ) ; and 3 < 4 , with m ∗ ( 3 ) = 2 < 7 = m ∗ ( 4 ) . Thus, the negation of the given statement is not true, and this completes the proof.

Theorem 9.

Let k and n be natural numbers such that k < n , then k m ∗ ( k ) ≤ nm ∗ ( n ) .

Proof.

Suppose not. That is k m ∗ ( k ) ≤ nm ∗ ( n ) for all k < n .

Dividing both sides by k m ∗ ( n ) gives m ∗ ( k ) m ∗ ( n ) > n k , ∀ k < n .

This implies

m ∗ ( k ) > m ∗ ( n ) > m ∗ ( n), ∀ k < n , contradicting Lemma 2.

Theorem 10.

Let γ i ≥ 2 and γ i + 1 > 2 for i ≥ 0 be consecutive natural numbers such that 2 is γ i -Mulatu summable and is γ i + 1 -Mulatu summable, respectively, and let γ i and γ i + 1 be Mulatu summable.

Proof.

We use induction on i .

For i = 0 , γ 0 = 2 , γ 1 = 3 and γ 0 + γ 1 = 5 = M 2 . Assuming it holds for i = k and Theorem 8, we get γ k + 1 + γ k + 2 = 1 2 ( 2 γ k + 1 + 2 γ k + 2 ) , which is a Mulatu’s number.

Example: What is the successor to γ = 3 and γ = 54 in Theorem 10?

The answer is 14 and 225 respectively.

Relationship with the Golden Ratio

The golden ratio is defined by taking a line segment and dividing it into two parts: the longer part (L) and the shorter part (S). The ratio of L to S is the same as the ratio of the entire line segment to L. Letting this ratio x , leads x = 1 + 1 x , whose solution is the golden ratio, ϕ = 1.6180339887 … .

The golden ratio pops up in so many places, from mathematics to art and nature. It’s like a secret thread that ties together beauty and balance. People often find it aesthetically pleasing, whether it’s in a stunning painting, an elegant building, or the way a sunflower blooms. ^{7,
9}

Furthermore, what’s particularly cool is that when we simplify the reciprocal of ϕ , it turns out to be just one less than ϕ itself. This means ϕ − 1 ϕ = 1 . It’s quite special that ϕ and its reciprocal are two numbers for which both their difference and their product equal one.

Surprisingly, Mulatu numbers have a remarkable connection to the golden ratio. When we divide one Mulatu number by the one before it, the result gets closer and closer to ϕ as the numbers increase.

Some of such ratios are: M 6 M 5 = 1.6470588235 M 7 M 6 = 1.6071428571 M 8 M 7 = 1.6222222222 . . … M 25 M 24 = 1.6180339884 M 26 M 25 = 1.6180339889 M 27 M 26 = 1.6180339887 …

This implies that lim n → ∞ M n + 1 M n = ϕ .

Conversely, if Mulatu’s number is divided by the succeeding Mulatu number, the result is close to the reciprocal of ϕ . Again, the larger the two numbers used, the closer the result to the reciprocal of ϕ .

Generating Functions for the Mulatu’s Number

In this section, we give a generating function for the sequence { M n } n = 0 ∞ . Definition 3.

The series ∑ n = 0 ∞ M n is called Mulatu’s series.

Theorem 11.

Mulatu’s series is divergent.

Proof.

Using the ratio test, lim n → ∞ M n + 1 M n = 1.618 > 1 . Thus, it diverges.

Theorem 12.

f ( x ) = ∑ n = 0 ∞ M n x n = 4 − 3 x 1 − x − x 2 is a generating function for the Mulatu’s sequence { M n } n = 0 ∞ .

Proof.

Let f ( x ) = ∑ n = 0 ∞ M n x n .

Claim: f ( x ) = 4 − 3 x 1 − x − x 2

Using the properties of Mulatu’s numbers, specifically, M n + 2 = M n + M n + 1 , we have f ( x ) = ∑ n = 0 ∞ M n + 2 x n = ∑ n = 0 ∞ M n + 1 x n + ∑ n = 0 ∞ M n x n = ∑ n = 1 ∞ M n x n − 1 + f ( x )

This implies, ∑ n = 2 ∞ M n x n = x ∑ n = 1 ∞ M n x n + x 2 f ( x )

That is − M 0 − M 1 x + f ( x ) = − M 0 x + xf ( x ) + x 2 f ( x ) .

Substituting M 0 and M 1 completes the proof.

Conclusion

In this work, we found many novel characterizations of Mulatu numbers. We produced relationships among Mulatu’s, Fibonacci’s, and Lucas’s numbers and with the set of natural numbers as well. Namely, we categorized two natural numbers or a natural number as the Mulatu summable or γ -Mulatu summable, respectively. Moreover, we have also shown that, similar to Fibonacci’s numbers, Mulatu’s numbers are related to the so-called golden ratio, which is most applicable in numerical optimization. Finally, we provide a generating function for the Mulatu numbers. These findings play a crucial role in both theoretical and applied mathematics.

Ethical declaration

We hereby declare that the information provided above is accurate, and that this research was conducted in compliance with all applicable ethical guidelines and institutional policies. Since this study did not involve any human or animal participants, there was no need for ethical approval or consent.

Declaration of originality

We declare that the research presented in this paper is our original work. This work has not been submitted for any other degree or qualification, and all the sources and references used have been appropriately acknowledged.

Data availability statement Underlying data

No data are associated with this article.

Extended data

Since it is a study on a mathematical theory, there is no extended data used.

References 1

Lemma

Lambright

Andreou

: The Mathematical Magic the Mulatu Numbers. Arts humanities hawaii. 2012; pp.1–11. Reference Source

Lemma

Tanksley

Brown

: The Mulatu Numbers in Actions. GPH-International Journal of Mathematics. 2021;4(02):11–16.

Rosen

: Elementary number theory. Pearson Education London;2011.

Jones

: Elementary number theory. Springer Science & Business Media;1998.

Lemma

Lambright

Atena

: Some Fascinating Theorems Of The Mulatu Numbers. Advances and Applications in Mathematical Sciences. 2016;15(4):133–138.

Mulatu

: The Amazing Mathematical Beauty of the Mulatu Numbers With Interesting Open Questions. IJRDO - Journal of Mathematics. 2018;4(4):01–10.

Koshy

: Fibonacci and Lucas numbers with applications. A Wiley-Interscience Publication;2001.

Knott

Quinney

Maths

: The life and numbers of Fibonacci. 1997;1:2007. Retrieved February.

Křížek

Somer

Šolcová

: Fibonacci and Lucas numbers. From Great Discoveries in Number Theory to Applications. 2021; pp.151–181. 10.1007/978-3-030-83899-7_7

Eaton

Bateman

Hauberg

: GNU Octave version 4.0. 0 manual: a high-level interactive language for numerical computations. 2015. 2019;8:13. Reference Source

Kiefer

: Sequential minimax search for a maximum. Proceedings of the American Mathematical Society. 1953;4(3):502–506. 10.1090/S0002-9939-1953-0055639-3

10.5256/f1000research.178948.r407779

Reviewer response for version 2

TAŞDEMİR

FUNDA

1 Referee 1Yozgat Bozok University, Yozgat, Turkey

Competing interests: No competing interests were disclosed.

18 9 2025

2025

This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

recommendation

approve-with-reservations

In this paper, the authors investigate Mulatu’s sequence, a type of linear recurrence sequence, with an emphasis on its basic algebraic properties.

The paper is easy to understand manner. While the identities presented are a useful starting point, they are not sufficient to fully characterize the sequence. I recommend that the authors include additional properties or identities to strengthen the mathematical foundation of the paper.

The paper should be well developed. It is not appropriate to index it in its current form. In my opinion, the paper could be accepted for indexing after a major revision according to the following suggestions:

The introduction needs to be expanded by situating the study within the context of relevant literature and emphasizing its significance and contributions.

The manuscript would benefit from replacing informal terms like “fascinating” with more precise and objective language to maintain an academic tone.

A more thorough and detailed exposition of the connection between Mulatu numbers, Fibonacci numbers, and the golden ratio is necessary to strengthen the mathematical foundations of the study.

In general, the theorems and proofs should be presented in a more formal and mathematically rigorous manner.

The manuscript refers to GNU Octave for numerical verification, yet lacks corresponding results. Presenting numerical evidence would reinforce the validity of the findings.

Page 4, proof of Lemma 1, should be “For n=0, .”

The identities/properties used should be either explicitly stated prior to their use or cited from relevant literature. (For example: Page 4, proof of the Theorem 1)

References should be improved and organized. (For example: Reference 1 (Lemma M,…) and Reference 6 (Mulatu L:)).

To improve the logical flow, intermediate computational steps should be explicitly presented. (For instance: Page 6, proof of the Theorem 8)

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Number theory, special sequences

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

ADMASU

AGEZE

Mathematics, Woldia University, Woldia, Amhara, Ethiopia

Competing interests: There is no competing interest.

22 9 2025

DEAR PROFESSOR FUNDA TAŞDEMİR

We are in the process of addressing the comments given and will submit the revised manuscript as soon as possible.

However, we would like to clarify that the GNU Octave were conducted to assist in the conceptualization and formulation of the conjectures presented in the paper, not for analysis or illustration of results.

Sincerely,

Ageze Abye Admasu

Woldia University, Ethiopia

10.5256/f1000research.178948.r405228

Reviewer response for version 2

Akkuş

Hakan

1 Referee https://orcid.org/0000-0001-9716-9424 1Erzincan Binali Yıldırım University, Erzincan, Turkey

Competing interests: No competing interests were disclosed.

5 9 2025

2025

recommendation

approve-with-reservations

Dear Editor,

I read the article from beginning to end. I can say that the links in the article are nice and clear. Furthermore, the arrangements are technically correct and have a logical structure with a beginning, middle, and end. I checked the procedures, and everything appears to be correct. I hope it will be of interest to the journal's readers. For the reasons mentioned above, I would recommend that you accept the article for the journal. However, I recommend that authors consider the following revisions before the Editor makes a decision:

*** I recommend including some previous work in the abstract, such as the generating functions and sum formulas of mulatu numbers.

*** Mulatu numbers are a special case of Fibonacci numbers, and I believe that mentioning Fibonacci numbers and their applications in the introduction will enhance the main message of the article and motivate readers.

*** I recommend reconsidering the definitions, theorems, and auxiliary theorems.

*** I find the references insufficient and recommend the addition of the following works.

Recommendation: Minor revision

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

I cannot comment. A qualified statistician is required.

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Number theory, Algebra, Applied mathematic

References 1

: Analytical Characterization of Self-Similarity in k-Cullen Sequences Through Generating Functions and Fibonacci Scaling. Fractal and Fractional .2025;9(6) : 10.3390/fractalfract9060380

10.3390/fractalfract9060380

: Properties of Generalized Bronze Fibonacci Sequences and Their Hyperbolic Quaternions. Axioms .2024;14(1) : 10.3390/axioms14010014

10.3390/axioms14010014

: Copper ratio obtained by generalizing the Fibonacci sequence. AIP Advances .2024;14(7) : 10.1063/5.0207147

10.1063/5.0207147

: Fibonacci and Lucas Numbers with Applications.2001; 10.1002/9781118033067

10.1002/9781118033067

: On the Fibonacci k-numbers. Chaos, Solitons & Fractals .2007;32(5) : 10.1016/j.chaos.2006.09.022 1615-1624

10.1016/j.chaos.2006.09.022

: On Thabit and Williams numbers base b as sum or difference of Fibonacci and Mulatu numbers and vice versa. Afrika Matematika .2025;36(1) : 10.1007/s13370-025-01266-0

10.1007/s13370-025-01266-0

: Mulatu Numbers as Products of Three Generalized Lucas Numbers. Mathematica Pannonica .2024;30_NS4(2) : 10.1556/314.2024.00014 142-161

10.1556/314.2024.00014

ADMASU

AGEZE

Mathematics, Woldia University, Woldia, Amhara, Ethiopia

Competing interests: There is no competing interest.

6 9 2025

Dear Professor Hakan Akkus,

Thank you for your valuable feedback on our manuscript, “More on the Fascinating Characterizations of Mulatu’s Numbers”. We have carefully considered all of your comments and suggestions. We agree with your assessment and will revise our manuscript in accordance with your recommendations.

We believe that these revisions will significantly improve the quality and clarity of our paper. We will submit the revised version as soon as it is complete.

Thank you again for your time and expertise.

Sincerely,

Ageze Abye Admasu

Department of Mathematics, Woldia University, Ethiopia

10.5256/f1000research.178948.r382531

Reviewer response for version 2

Adegoke

Kunle

1 Referee 1Obafemi Awolowo University, Ife, Osun, Nigeria

Competing interests: No competing interests were disclosed.

9 6 2025

2025

recommendation

reject

The authors present results for a particular case of a well-known sequence of numbers, the generalized Fibonacci sequence, also called the Gibonacci sequence. Since all the results can be obtained immediately by substituting the starting values G0=4 and G1=1 in the known general results, the results in this manuscript are trivial. I do not recommend indexing in F1000Research.

Is the work clearly and accurately presented and does it cite the current literature?

Yes

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

I confirm that I have read this submission and believe that I have an appropriate level of expertise to state that I do not consider it to be of an acceptable scientific standard, for reasons outlined above.

10.5256/f1000research.178948.r382539

Reviewer response for version 2

Duman

Merve Güney

1 Referee 1Sakarya University of Applied Sciences, Sakarya, Turkey

Competing interests: No competing interests were disclosed.

2 6 2025

2025

recommendation

reject

In this manuscript, some properties of Mulatu’s numbers are given. Many of these properties are either known or have obvious proofs.

There are too many deficiencies in the presented manuscript.

Proof methods were chosen incorrectly.

Very simple identities were given.

Some proofs were not given.

There are format errors.

The presentation style of the manuscript is not understandable.

Introduction should be expanded.

References should be improved.

Some intermediate operations should be given.

There are too many spelling and format errors.

The article must be well developed. It is not appropriate to publish it in its current form.

Is the work clearly and accurately presented and does it cite the current literature?

If applicable, is the statistical analysis and its interpretation appropriate?

Partly

Are all the source data underlying the results available to ensure full reproducibility?

Partly

Is the study design appropriate and is the work technically sound?

Are the conclusions drawn adequately supported by the results?

Are sufficient details of methods and analysis provided to allow replication by others?

Partly

Reviewer Expertise:

Number theory, Diophantine equations, Special sequences, etc.

10.5256/f1000research.178948.r382535

Reviewer response for version 2

Hashim

Hayder R.

1 Referee https://orcid.org/0000-0001-5408-7496 1University of Kufa, Kufa, Najaf Governorate, Iraq

Competing interests: No competing interests were disclosed.

28 5 2025

2025

recommendation

approve-with-reservations

This article mainly gives nice results to a linear recurrence sequence called Mulatu’s sequence.

The article is well written, but I suggest to improve it as follows:

- This sequence is a linear recurrence sequence, so you should recall the concept of this type of such sequences regarding to their definition, kinds, and properties.

- In the abstract, you mentioned that the numbers of this sequences represent revolutionary contributions to the mathematical world. What are these contributions to the mathematical world? Can you recall some of the applications to this sequence? Why is it important? Please improve the introduction with the answers of those questions.

- In the proof of Lemma 1:" For n=0, gcd(M0,M1)=gcd(41)=1." should be "For n=0, gcd(M0,M1)=gcd(4,1)=1."

- In Theorem 1"Adding any ten consecutive Mulatu numbers together will always result in a number that is divisible by 11." Can you provide examples satisfying the theorem, and not satisfying the theorem in case of adding more or less that 10 consecutive Mulatu numbers?

-Relationship with the Golden Ratio. You supposed that limit of M_{n+1}/M_{n}= 1.618...., can you provide a proof for it.

Is the work clearly and accurately presented and does it cite the current literature?

Partly

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

No source data required

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Yes

Reviewer Expertise:

Number theory

ADMASU

AGEZE

Mathematics, Woldia University, Woldia, Amhara, Ethiopia

Competing interests: There is no competing interest.

22 9 2025

Dear Professor Hayder R. Hashim

I would like to extend my sincere gratitude for your insightful comments and valuable feedback on my article. Your suggestions have been incredibly helpful in improving the quality of our work. I truly appreciate the time and effort you dedicated to reviewing it. We will submit the revised manuscript as soon as possible.

Sincerly,

Ageze Abye Admasu

Woldia University, Ethiopia

10.5256/f1000research.173234.r358140

Reviewer response for version 1

Mandal

Priyabrata

1 Referee https://orcid.org/0000-0001-6472-6239 1Manipal Institute of Technology, Manipal, India

Competing interests: No competing interests were disclosed.

17 2 2025

2025

recommendation

approve-with-reservations

Title and Abstract:

The title is appropriate and reflects the paper's content.

The abstract is clear but somewhat not standard. The phrase "beautiful and incredible patterns" is informal for an academic paper. A more precise statement would improve clarity.

Introduction:

The introduction does not adequately define the novelty of the work. The discussion on the relationship between Mulatu numbers, Fibonacci numbers, and the golden ratio should be more structured.

Mathematical Rigor and Errors:

Lemma 1 states, "Any two consecutive Mulatu numbers are relatively prime." A proof or reference is missing.

In the proof of Theorem 1, the table for j,x, and y is completely unnecessary. The author should identify explicitly what those x and y actually mean. By this, I mean to show that M _i+j=F _j-1M _i + F _j M _i+1, where F _n denotes the nth Fibonacci number.

The proof of Theorem 2 contains an induction step that is not fully justified. There is no need for induction here; just use the recurrence relation for M _n.

In Theorem 3, the proof is poorly structured, making it difficult to follow the logic behind the claim. No need for induction; note that M _2n=(M _2n+1- M _2n-1), and proceed.

Several theorems, such as Theorems 6 and 7, rely on induction. I expect direct proof rather than induction.

Theorem 8 lacks a rigorous explanation of how the divisibility property is maintained across different indices.

Computational Verification:

The paper states that GNU Octave was used for numerical verification but does not present any computational results. A section summarizing numerical experiments would strengthen the claims.

The relationship with the golden ratio is interesting, but the claim that "Mulatu numbers can be used to calculate the golden ratio" should be better substantiated with precise mathematical arguments.

Language and Formatting Issues:

Several grammatical errors appear throughout the text, such as:

"These findings play a vital role in the boarder context of mathematical sequences" (should be "broader").

"We have also shown that, similar to Fibonacci’s numbers, Mulatu’s numbers also give the so-called golden ratio" (redundant use of "also").

The paper should maintain a more formal tone and avoid phrases like "fascinating patterns" without proper justification.

The formatting of mathematical expressions is inconsistent, particularly in the placement of summations and recurrence relations.

References:

Several references lack proper formatting. Please organize them in a certain way.

Overall Recommendation: While the paper introduces interesting results related to Mulatu numbers, there are significant issues with the rigour of proofs, mathematical notation, and clarity of exposition. The authors should:

Strengthen the proofs with clearer justifications and better-structured induction arguments.

Provide numerical verification where computational methods are referenced.

Improve the language and formatting to enhance readability.

Clarify the motivation behind the new definitions and properties introduced.

After these major revisions, the paper would be more suitable for peer review and indexing.

Is the work clearly and accurately presented and does it cite the current literature?

If applicable, is the statistical analysis and its interpretation appropriate?

Not applicable

Are all the source data underlying the results available to ensure full reproducibility?

Yes

Is the study design appropriate and is the work technically sound?

Yes

Are the conclusions drawn adequately supported by the results?

Yes

Are sufficient details of methods and analysis provided to allow replication by others?

Reviewer Expertise:

Number Theory

ADMASU

AGEZE

Mathematics, Woldia University, Woldia, Amhara, Ethiopia

Competing interests: No competing interest.

3 3 2025

Dear Professor Priyabrata Mandal,

I hope this message finds you well. I wanted to take a moment to express my sincere gratitude for the time and effort you dedicated to reviewing my article, More on the fascinating characterizations of Mulatu’s numbers . Your thoughtful and constructive comments have been incredibly valuable in improving the quality of the work.

I particularly appreciate the insights you provided deeply. Your feedback helped me see new perspectives and refine key sections of the manuscript. The suggestions you offered were both insightful and thorough, and I am confident that the article has greatly benefitted from your expertise.

Once again, thank you for your careful review and constructive input. Your contributions are highly valued, and I am grateful for the support you’ve given in helping me improve my work.

With regards,

Ageze Abye Admasu

Ageze.ab19@gmail.com

Point-by-point response

Manuscript title: More on the fascinating characterizations of Mulatu’s numbers

DOI: 10.12688/f1000research.157738.1

First of all, we want to thank the reviewers and editors sincerely for their insightful and encouraging comments.

Reviewer Comments and our response

Title and Abstract

Reviewer Comments

•The title is appropriate and reflects the paper's content.

•The abstract is clear but somewhat not standard. The phrase "beautiful and incredible patterns" is informal for an academic paper. A more precise statement would improve clarity.

Our Response

Thank you, and sorry for the inconvenience

All comments given here are accepted and corrected.

Introduction:

Reviewer Comments

The introduction does not adequately define the novelty of the work. The discussion on the relationship between Mulatu numbers, Fibonacci numbers, and the golden ratio should be more structured.

Our Response

Thank you, we modified it.

Mathematical Rigor and Errors:

Reviewer Comments

Lemma 1 states, "Any two consecutive Mulatu numbers are relatively prime." A proof or reference is missing.

The proof of Theorem 2 contains an induction step that is not fully justified. There is no need for induction here; just use the recurrence relation for M _n.

In Theorem 3, the proof is poorly structured, making it difficult to follow the logic behind the claim. No need for induction; note that M _2n=(M _2n+1- M _2n-1), and proceed.

Several theorems, such as Theorems 6 and 7, rely on induction. I expect direct proof rather than induction.

Theorem 8 lacks a rigorous explanation of how the divisibility property is maintained across different indices.

Our Response

Noted!

Accordingly, we try to address each comment on the revised manuscript as follows:

Lemma 1 is proved.

Theorem 1, 2 and 3 are proved using the comments.

Apricating your view, the proofs of theorem 6 and 7 using recursion are relatively longer to publish on this manuscript and it will not compromise the quality of the paper.

Theorem 8 is clarified.

Computational Verification

Reviewer Comments

The paper states that GNU Octave was used for numerical verification but does not present any computational results. A section summarizing numerical experiments would strengthen the claims.

Our Response

We appreciate your view. However, the use of GNU Octave was only to conceptualize hypothesis of theorems by taking numerical examples before formal proofs are done. We have also revised the manuscript to state this clearly.

Reviewer Comments

Our Response

We apologize for the inconvenience. It is edited.

Language and Formatting Issues

Reviewer Comments

Several grammatical errors appear throughout the text, such as:

"These findings play a vital role in the boarder context of mathematical sequences" (should be "broader").

"We have also shown that, similar to Fibonacci’s numbers, Mulatu’s numbers also give the so-called golden ratio" (redundant use of "also").

The paper should maintain a more formal tone and avoid phrases like "fascinating patterns" without proper justification.

The formatting of mathematical expressions is inconsistent, particularly in the placement of summations and recurrence relations.

Our Response

Thank you, and sorry for your inconvenience

All comments given here are accepted and corrected.

References

Reviewer Comments

Several references lack proper formatting. Please organize them in a certain way.

Our Response

Thank you, and sorry for the inconvenience. We have revised references on the manuscript accordingly.

Overall Recommendation

Reviewer Comments

While the paper introduces interesting results related to Mulatu numbers, there are significant issues with the rigour of proofs, mathematical notation, and clarity of exposition. The authors should:

Strengthen the proofs with clearer justifications and better-structured induction arguments.

Provide numerical verification where computational methods are referenced.

Improve the language and formatting to enhance readability.

Clarify the motivation behind the new definitions and properties introduced.

Our Response

All comments given here are accepted and corrected accordingly.

Thank you, reviewers!!

We got many constructive comments and suggestions which helps to further development of the manuscript and experience for our future work.