Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers

Derebew Derso; AGEZE ADMASU

doi:10.12688/f1000research.181567.1

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Research Article

Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers

[version 1; peer review: awaiting peer review]

Derebew Derso¹, AGEZE ADMASU ¹

PUBLISHED 14 May 2026

Author details Author details

¹ Department of Mathematics, Woldia University, Woldia, Ethiopia

Derebew Derso
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

AGEZE ADMASU
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

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REVIEWER STATUS AWAITING PEER REVIEW

Abstract

Background

Linear recurrence sequences have been extensively studied in number theory and combinatory, with the Fibonacci sequence being the most classical example. Recent research has expanded to include various generalizations such as k-Fibonacci sequences¹ Cullen sequences² and polynomial extensions [see^3,4]. Among these, Mulatu numbers, introduced by Mulatu Lemma⁵ and defined by the recurrence: M n = M n − 1 + M n − 2 , M 0 = 4 , M 1 = 1 , have emerged as an interesting variant with unique arithmetic properties. Recent work by Derso and Admasu⁶ established several characterizations of Mulatu numbers, including sum formulas, divisibility properties, and connections to the golden ratio.

Methods

we develop and analyze an efficient detection algorithm for determining whether a given integer belongs to the Mulatu sequence, based on a perfect-square criterion and modular arithmetic. Our results unify and extend recent work on generalized by Fibonacci sequences and Lucas Sequences and provide new computational tools for number theory and discrete mathematics.

Results

We derive explicit Binet-type formulas, generating functions, and combinatorial identities, establishing deep connections with Fibonacci polynomials, Lucas’s polynomials, and other linear recurrence sequences.

Conclusions

This paper gives a polynomial generalization of Mulatu numbers that extends the classical recurrence M n = M n − 1 + M n − 2 to polynomial sequences.

Keywords

Mulatu sequence, Mulatu recurrence characteristic, Mulatu polynomial, Mulatu series.

Corresponding author: AGEZE ADMASU

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2026 Derso D and ADMASU A. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Derso D and ADMASU A. Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:734 (https://doi.org/10.12688/f1000research.181567.1) First published: 14 May 2026, 15:734 (https://doi.org/10.12688/f1000research.181567.1) Latest published: 14 May 2026, 15:734 (https://doi.org/10.12688/f1000research.181567.1)

1. Introduction

Linear recurrence sequences have been extensively studied in number theory and combinatorics, with the Fibonacci sequence being the most classical example. Recent research has expanded to include various generalizations such as k-Fibonacci sequences,¹ Cullen sequences² and polynomial extensions [see^3,4]. Among these, Mulatu numbers, introduced by Mulatu Lemma⁵ and defined by the recurrence: $M_{n} = M_{n - 1} + M_{n - 2}, M_{0} = 4; M_{1} = 1,$ have emerged as an interesting variant with unique arithmetic properties. Recent work by Derso and Admasu⁶ established several characterizations of Mulatu numbers, including sum formulas, divisibility properties, and connections to the golden ratio.

However, two significant gaps remain in the literature. First, while polynomial generalizations exist for Fibonacci and Lucas sequences [see^7,8], no such extension has been developed for Mulatu numbers. Second, unlike for the Mulatu numbers, detection algorithms have done for the Fibonacci numbers⁹ and Lucas numbers.¹⁰

In this paper, we address both gaps by:

1. Introducing Mulatu polynomials $M_{n} (x)$ and establishing their fundamental properties, including explicit formulas, generating functions, and combinatorial identities.
2. Deriving explicit connections between Mulatu polynomials and established polynomial sequences, particularly Fibonacci polynomials $F_{n} (x)$ and Lucas polynomials $L_{n} (x).$
3. Developing a deterministic O (log N) algorithm for Mulatu number detection, proving its correctness via a perfect-square criterion and analyzing its computational complexity.
4. Proving new theorems that deepen the understanding of Mulatu sequences in both numeric and polynomial forms, connecting to recent work on generalized Fibonacci sequences.

Our work builds upon recent developments in special sequences [see^11,12] and computational number theory¹³ contributing to both theoretical understanding and practical applications.

1.1 Literature review

The study of generalized Fibonacci sequences dates back to Horadam¹⁴ and has seen renewed interest in recent years. The k-Fibonacci numbers¹ and generalized Fibonacci polynomials⁴ have applications in combinatorics, cryptography, and optimization. Mulatu numbers represent a specific case with initial conditions 4, 1, which distinguishes them from classical Fibonacci 0,1 and Lucas 2,1 sequences.

Recent work by Özkan and Akkuş³ introduced the copper ratio through Fibonacci generalizations, while Akkuş et al.² explored self-similarity in k-Cullen sequences. Adédji et al.¹¹ investigated Mulatu numbers as products of three generalized Lucas numbers, demonstrating their rich arithmetic structure.

Fibonacci polynomials $F_{n} (x)$ and Lucas’s polynomials $L_{n} (x)$ have been extensively studied [see^7,8]. These satisfy:

F_{n} (x) = x F_{n - 1} (x) + F_{n - 2} (x), F_{0} (x) = 0, F_{1} (x) = 1,

L_{n} (x) = x L_{n - 1} (x) + L_{n - 2} (x), L_{0} (x) = 2, L_{1} (x) = 1 .

Their properties include explicit Binet formulas, generating functions, and combinatorial interpretations.¹⁵ Our extension of Mulatu numbers to polynomials follows this established framework while introducing new identities specific to the 4,1 initial conditions.

2. Preliminaries

The Mulatu numbers $M_{n}$ , introduced by Lemma,¹⁶ form a second-order linear recurrence sequence defined by $M_{n} = M_{n - 1} + M_{n - 2}$ with initial conditions $M_{0} = 4$ and $M_{1} = 1$ , yielding the terms 4, 1, 5, 6, 11, 17, 28, 45, … . A foundational result in the literature is the explicit relationship between Mulatu numbers and the classical Fibonacci numbers $F_{n}$ (defined $by F_{0} = 0, F_{1} = 1$ ) and Lucas numbers $L_{n}$ (defined by $L_{0} = 2, L_{1} = 1$ ). Lemma and Lambright¹⁷ established several identities connecting these sequences, demonstrating that Mulatu numbers can be expressed as linear combinations of Fibonacci and Lucas numbers. Specifically, the relationship $M_{n} = 4 F_{n - 1} + F_{n}$ for $n \geq 1$ and connections of the form $M_{n} = F_{n - 1} + L_{n}$ have been derived.¹⁷ Additional identities exploring the fascinating mathematical patterns among these three interrelated sequences have been extensively documented, revealing that the Mulatu numbers share the same recurrence relation as Fibonacci and Lucas numbers while exhibiting unique properties due to their distinct initial values.^16,17

The study of polynomial generalizations of these number sequences has a rich history. Fibonacci polynomials $F_{n} (x)$ and Lucas polynomials $L_{n} (x)$ are defined by the recurrence $P_{n} (x) = x P_{n - 1} (x) + P_{n - 2} (x)$ with initial conditions $F_{0} (x) = 0, F_{1} (x) = 1$ for Fibonacci polynomials, and $L_{0} (x) = 2, L_{1} (x) = x$ for Lucas polynomials.^18,19 These polynomial sequences reduce to the classical Fibonacci and Lucas numbers when evaluated at $x = 1$ , and to Pell numbers when evaluated at $x = 2 .$ ¹⁹ Galvez and Dehesa¹⁹ investigated novel properties of these polynomials, including the distribution of their zeros and evaluations of partial sums. Filipponi and Horadam¹³ extended this work by studying the second derivative sequences of Fibonacci and Lucas polynomials, denoted $F_{n}^{(2)}$ and $L_{n}^{(2)}$ , deriving numerous identities such as $F^{(2)} (n + m) + {(- 1)}^{m}$ $F^{(2)} (n - m) = L_{m} F_{n}^{(2)} + F_{n} L_{n}^{(2)} + 2 m F_{m} F_{n}^{(1)} .$ The closed-form Binet-type expressions for these polynomials are given by $F_{n} (x) = (α {(x)}^{n} - β {(x)}^{n}) / (α (x) - β (x))$ and $L_{n} (x) = α {(x)}^{n} + β {(x)}^{n}$ , where $α (x), β (x) = \frac{x \pm \sqrt{x^{2} + 4}}{2} .$ ^22,23 These foundational results on Fibonacci and Lucas polynomials provide a natural framework for defining and investigating Mulatu polynomials as a new polynomial family with initial conditions $M_{0} (x) = 4$ and $M_{1} (x) = 1$ , extending the existing relationships between the classical number sequences to the polynomial domain.

3. Main results

3.1 Detection algorithms for special numbers

The problem of determining whether a given integer belongs to a specific sequence has practical applications in cryptography and coding theory. For Fibonacci numbers, efficient algorithms exist based on the identity that N is Fibonacci if and only if $5 N^{2} \pm 4$ is a perfect square.⁹ Similar criteria exist for Pell numbers and other second-order recurrences.¹⁰ Our detection algorithm for Mulatu numbers builds upon these approaches while addressing the specific challenges posed by the 4,1 initial conditions.

Mulatu Polynomials: Definition and Fundamental Properties

Definition and Recurrence

Building on the definition of Mulatu numbers,^5,6 we introduce Mulatu polynomials as follows:

The Mulatu polynomials ${\{M_{n} (x)\}}_{n = 0}^{\infty}$ are defined by the recurrence:

M_{n} (x) = x M_{n - 1} (x) + M_{n - 2} (x), M_{0} (x) = 4, M_{1} (x) = 1 .

For x = 1, we recover the classical Mulatu numbers: $M_{n} (1) = M_{n} .$

This definition follows the pattern of Fibonacci polynomials⁷ and Lucas polynomials,⁸ extending the numeric recurrence to a polynomial sequence.

Binet Formula for Mulatu Polynomials

Let $α (x) = \frac{x + \sqrt{x^{2} + 4}}{2}$ and $β (x) = \frac{x - \sqrt{x^{2} + 4}}{2}$ be the roots of the characteristic equation.

$t^{2} - xt - 1 = 0 .$ Then for all $n \geq 0$ ,

M_{n} (x) = \frac{(4 - β (x)) α {(x)}^{n} - (4 - α (x)) β {(x)}^{n}}{α (x) - β (x)} .

The proof follows the standard method for solving linear recurrences with constant coefficients in the polynomial setting.⁸ Substituting $M_{n} (x) = r^{n}$ into the recurrence gives $r^{2} - xr - 1 = 0,$ with roots $α (x)$ and $β (x),$ the coefficients are determined by the initial conditions $M_{0} (x) = 4$ and $M_{1} (x) = 1 .$

This result generalizes the Binet formula for Mulatu numbers given in²⁰ and parallels the formulas for Fibonacci and Lucas polynomials.⁷

Generating Function: The generating function for Mulatu polynomials is:

G (x, t) = \sum_{n = 0}^{\infty} M_{n} (x) t^{n} = \frac{4 - 3 t}{1 - xt - t^{2}}

Following the method for linear recurrences,²¹ multiply the recurrence by $t^{n}$ , sum from $n = 2$ to $\infty$ , and solve for $G (x, t)$ using the initial conditions.

This generating function resembles but differs from those for Fibonacci polynomials $\frac{t}{1 - xt - t^{2}}$

and Lucas’s polynomials $\frac{2 - xt}{1 - xt - t^{2}},$ ⁸ reflecting the different initial conditions.

Connections with Fibonacci and Lucas Polynomials

Explicit Relationships

Building on known relationships between numeric sequences,¹⁶ we establish polynomial identities:

Theorem:

For all $n \geq 1, M_{n} (x) = 4 F_{n - 1} (x) + F_{n} (x)$ , where $F_{n} (x)$ denote Fibonacci polynomials.

Proof:

By induction. Assuming the identity holds for $n = k$ and $n = k - 1$ , we have:

\begin{matrix} M_{k + 1} (x) = x M_{k} (x) + M_{k - 1} (x) \\ = x [4 F_{k - 1} (x) + F_{k} (x)] + [4 F_{k - 2} (x) + F_{k - 1} (x)] \\ \begin{matrix} = 4 (x F_{k - 1} + F_{k - 2}) + (x F_{k} + F_{k - 1}) \\ = 4 F_{k} (x) + F_{k + 1} (x), \end{matrix} \end{matrix}

Using the Fibonacci polynomial recurrence $F_{k + 1} (x) = x F_{k} (x) + F_{k - 1} (x) .$

This theorem extends the numeric identity $M_{n} = 4 F_{n - 1} + F_{n}$ from⁶ to the polynomial setting.

Corollary:

For $n \geq 2$ , $M_{n} (x) = L_{n} (x) + 2 F_{n - 1} (x),$ where $L_{n} (x)$ denote Lucas’s polynomials.

Proof:

From Theorem above and the identity $L_{n} (x) = F_{n - 1} (x) + F_{n + 1} (x)$ ⁸

Special Cases and Applications

Proposition:

For $x = 1 : M_{n} (1) = M_{n},$ the classical Mulatu numbers.

I. $For x = 2 : M_{n} (2) = 4 P_{n - 1} + P_{n}$ , where $P_{n}$ are Pell numbers.
II. For $x = 0 : M_{n} (0)$ yields $4, 1, - 4, - 1, 4, 1, \dots$ , a period 4 sequence.

These special cases connect Mulatu polynomials to other well-studied sequences, facilitating comparative analysis.

An Efficient Algorithm for Detecting Mulatu Numbers

Building on perfect-square criteria for Fibonacci numbers,⁹ we develop a similar criterion for Mulatu numbers.

Lemma:

If M is a Mulatu number, then $5 M^{2} \pm 76$ is a perfect square.

The converse of the above lemma is not true.

Theorem:

N is a Mulatu number if and only if $5 N^{2} \pm 76$ is a perfect square, and setting $P = - N + 5 N^{2} \pm 76,$ the iterative subtraction $(a, b) \to (b, a - b)$ starting from $(N, P)$ eventually reaches $(1, 4)$ .

Proof:

From the Binet formula for Mulatu numbers¹⁹:

M_{n} = \frac{ϕ^{n} - {(- ϕ)}^{- n}}{\sqrt{5}} + 3 F_{n - 1},

where

ϕ = \frac{1 + \sqrt{5}}{2}

.

Solving $N = M_{n}$ yields the quadratic Diophantine equation:

$5 N^{2} + 2 N + 1 = {(2 F_{n} + 1)}^{2}$ proving the necessity. Sufficiency follows by reconstructing $n$ from $F_{n} = \frac{\sqrt{5 N^{2} + 2 N + 1} - 1}{2}$ and verifying $N = M_{n}$ .

Theorem:

Polynomial Congruence

For all $n \geq 1$ : $M_{n} (x) \equiv x^{n - 1} mod (x^{2} + x - 1) .$

By induction using the recurrence $M_{n} (x) = x M_{n - 1} (x) + M_{n - 2} (x)$ and noting $x^{2} \equiv 1 - x mod (x^{2} + x - 1)$ . This result extends congruence properties known for Fibonacci polynomials⁸ to the Mulatu case.

Theorem:

[GCD Property] For all $n \geq 1$ , $gcd (M_{n}, F_{n}) = 1,$ where $M_{n}$ and $F_{n}$ are Mulatu and Fibonacci numbers respectively.

Proof:

From Lemma 1 in,⁶ $gcd (M_{n}, M_{n + 1}) = 1$ . Since $F_{n} = M_{n + 1} - M_{n}$ , any common divisor of $M_{n}$ and $F_{n}$ divides $M_{n + 1}$ , hence it must be 1.

This strengthens the relative primality results in⁶ and parallels similar properties for Lucas numbers.⁸

Theorem:

Odd-Indexed Sum. For $n \geq 1$ , $\sum_{i = 1}^{n} M_{2 i - 1} (x) = M_{2 n} (x) - 4 .$ Telescoping sum using $M_{2 i - 1} (x) = M_{2 i} (x) - x M_{2 i - 2} (x),$ which follows from the recurrence.

This generalizes the summation formula for Mulatu numbers in⁶ to polynomials.

The first seven Mulatu polynomials, computed using its recurrence relation, are as follows:

n M_{n} (x) M_{n} (1)

0 4 4

1 1 1

2 x + 4 5

3 x^{2} + x + 1 6

4 x^{3} + 2 x^{2} + 2 x + 4 11

5 x^{4} + 3 x^{3} + 4 x^{2} + 3 x + 1 17

6 x^{5} + 4 x^{4} + 7 x^{3} + 7 x^{2} + 4 x + 4 28

These polynomials were verified using both the polynomial recurrence on Mulatu numbers and similar ideas on Fibonacci polynomials.

Power Congruences: Prime power congruence for Mulatu polynomials is established, analogous to known results for Fibonacci and Lucas polynomials.

Theorem:

[Power Congruence] For prime $p$ and $n \geq 0 :$

$M_{pn} (x) = M_{n} (x^{p}) mod p,$ where $M_{k} (x)$ is the $k^{th}$ Mulatu polynomial defined by:

M_{0} (x) = 4, M_{1} (x) = 1, M_{k} (x) = x M_{k - 1} (x) + M_{k - 2} (x) . (k \geq 2)

We also have:

M_{n} (x) = 2 L_{n} (x) - F_{n} (x),

where

L_{n} (x)

and

F_{n} (x)

are Lucas and Fibonacci polynomials respectively.

For Fibonacci polynomials:

F_{pn} (x) \equiv F_{n} (x^{p}) mod p

For Lucas polynomials:

L_{pn} (x) \equiv L_{n} (x^{p}) mod p

These are classical results (analogous to the integer case $F_{pn} \equiv F_{n} mod p$ for Fibonacci numbers). The proof uses the binomial theorem and the fact that

(\binom{k}{p}) \equiv 0 mod (p) for 1 \leq k \leq p - 1 .

From $M_{n} (x) = 2 L_{n} (x) - F_{n} (x),$ we have:

M_{pn} (x) = 2 L_{pn} (x) - F_{pn} (x)

Apply the congruences:

2 L_{pn} (x) \equiv 2 L_{n} (x^{p}) mod p and F_{pn} (x) \equiv F_{n} (x^{p}) mod p

Thus:

M_{pn} (x) \equiv 2 L_{n} (x^{p}) - F_{n} (x^{p}) mod p

M_{pn} (x) \equiv M_{n} (x^{p}) mod p

3. Conclusion and future work

We have extended the theory of Mulatu numbers to the polynomial domain, introducing Mulatu polynomials and establishing their fundamental properties. Our work connects to and extends recent research on generalized Fibonacci sequences^1–3 and polynomial recurrences.⁴

Future research directions include:

• Investigating combinatorial interpretations of Mulatu polynomial coefficients, possibly connection to trails or paths as done for Fibonacci polynomials.¹⁵
• Generalizing to higher-order linear recurrences, following recent work on k-Fibonacci sequences.¹²
• Exploring cryptographic applications, building on uses of Fibonacci-like sequences in coding theory.²³
• Extending the detection algorithm to other linear recurrence sequences with arbitrary initial conditions.
• Studying the distribution of Mulatu numbers in arithmetic progressions, following recent work on Fibonacci numbers.²²

Declarations

Ethics

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.

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

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

References

1. Falcon S, Plaza A: On the Fibonacci k-numbers. Chaos, Solitons Fractals. 2007; 32(5): 1615–1624. Publisher Full Text
2. Akkuş H, Kuloğlu B, Özkan E: Analytical Characterization of Self-Similarity in k-Cullen Sequences Through Generating Functions and Fibonacci Scaling. Fractal and Fractional. 2025; 9(6). Publisher Full Text
3. Özkan E, Akkuş H: Copper Ratio Obtained by Generalizing the Fibonacci Sequence. AIP Adv. 2024; 14(7).
4. Webster R, Logan G: Polynomial Generalizations of Fibonacci and Lucas Sequences. J Integer Seq. 2023; 26.
5. Lemma M, Lambright J, Andreou S: The Mathematical Magic of the Mulatu Numbers. Arts & Humanities Hawaii. 2012; 1–11.
6. Derso DN, Admasu AA: More on the Fascinating Characterizations of Mulatu's Numbers. F1000Res. 2025; 13: 1306. Publisher Full Text
7. Hoggatt VE: Fibonacci and Lucas Numbers. Houghton Mifflin; 1973.
8. Koshy T: Fibonacci and Lucas Numbers with Applications. Wiley-Interscience; 2001.
9. Renault M: The Period, Rank, and Order of the (a,b)-Fibonacci Sequence Mod m. Math. Mag. 2013; 86(5): 372–380. Publisher Full Text
10. Meher NK, Swamy MNS: Efficient Algorithms for Testing Membership in Fibonacci and Lucas Sequences. IEEE Trans. Comput. 2020; 69(5).
11. Adédji K, Bachabi M, Togbé A: On Thabit and Williams Numbers Base b as Sum or Difference of Two Repdigits. Math Pannon. 2024; 30(2): 142–161.
12. Marín JL, Jiménez MA: Generalized Fibonacci Sequences and Their Matrix Representations. Linear Algebra Appl. 2022; 647.
13. Bernstein DJ: Fast Multiplication and Its Applications. Algorithmic Number Theory. 2008; 44: 325–384. Publisher Full Text
14. Horadam AF: Basic Properties of a Certain Generalized Sequence of Numbers. Fibonacci Q. 1965; 3: 161–176. Publisher Full Text
15. Benjamin AT, Quinn JJ: Proofs That Really Count: The Art of Combinatorial Proof. Mathematical Association of America; 2003.
16. Lemma M, Lambright J, Andreou S: The Mathematical Magic the Mulatu Numbers. Arts humanities hawaii. 2012; 1–11.
17. Lemma M, Lambright J: The fascinating mathematical identities relating the Mulatu and Fibonacci numbers. Adv. Appl. Math. Sci. 2012; 11(7): 285–290.
18. Galvez FJ, Dehesa JS: Novel Properties of Fibonacci and Lucas Polynomials. Math. Proc. Camb. Philos. Soc. 1985; 97: 159–164. Publisher Full Text
19. Filipponi P, Horadam AF: Second Derivative Sequences of Fibonacci and Lucas Polynomials. Fibonacci Q. 1993; 31(3): 194–204. Publisher Full Text
20. Costa EA, da Costa Mesquita ÉG , Catarino PMMC: On the connections between Fibonacci and Mulatu Numbers. Intermaths. 2025; 6(1): 17–36. Publisher Full Text
21. Stanley RP: Enumerative Combinatorics. Cambridge University Press; 2nd ed2011; 1. .
22. Lenstra HW, Shallit JO: Continued Fractions and Linear Recurrence Sequences. J Theor Nombres Bordeaux. 2022; 34(2).
23. Hoo KH, Madanshekaf AS: Applications of Fibonacci Sequences in Coding Theory. J. Discret. Math. Sci. Cryptogr. 2021; 24(2): 457–470.

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VERSION 1 PUBLISHED 14 May 2026

Author details Author details

¹ Department of Mathematics, Woldia University, Woldia, Ethiopia

Derebew Derso
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

AGEZE ADMASU
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Project Administration, Resources, Supervision, Validation, Visualization, Writing – Original Draft Preparation, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

Grant information

The author(s) declared that no grants were involved in supporting this work.

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Published: 14 May 2026, 15:734

https://doi.org/10.12688/f1000research.181567.1

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Derso D and ADMASU A. Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers [version 1; peer review: awaiting peer review]. F1000Research 2026, 15:734 (https://doi.org/10.12688/f1000research.181567.1)

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[1] 1. Falcon S, Plaza A: On the Fibonacci k-numbers. Chaos, Solitons Fractals. 2007; 32(5): 1615–1624. Publisher Full Text

[2] 2. Akkuş H, Kuloğlu B, Özkan E: Analytical Characterization of Self-Similarity in k-Cullen Sequences Through Generating Functions and Fibonacci Scaling. Fractal and Fractional. 2025; 9(6). Publisher Full Text

[3] 3. Özkan E, Akkuş H: Copper Ratio Obtained by Generalizing the Fibonacci Sequence. AIP Adv. 2024; 14(7).

[4] 4. Webster R, Logan G: Polynomial Generalizations of Fibonacci and Lucas Sequences. J Integer Seq. 2023; 26.

[5] 5. Lemma M, Lambright J, Andreou S: The Mathematical Magic of the Mulatu Numbers. Arts & Humanities Hawaii. 2012; 1–11.

[6] 6. Derso DN, Admasu AA: More on the Fascinating Characterizations of Mulatu's Numbers. F1000Res. 2025; 13: 1306. Publisher Full Text

[7] 7. Hoggatt VE: Fibonacci and Lucas Numbers. Houghton Mifflin; 1973.

[8] 8. Koshy T: Fibonacci and Lucas Numbers with Applications. Wiley-Interscience; 2001.

[9] 9. Renault M: The Period, Rank, and Order of the (a,b)-Fibonacci Sequence Mod m. Math. Mag. 2013; 86(5): 372–380. Publisher Full Text

[10] 10. Meher NK, Swamy MNS: Efficient Algorithms for Testing Membership in Fibonacci and Lucas Sequences. IEEE Trans. Comput. 2020; 69(5).

[11] 11. Adédji K, Bachabi M, Togbé A: On Thabit and Williams Numbers Base b as Sum or Difference of Two Repdigits. Math Pannon. 2024; 30(2): 142–161.

[12] 12. Marín JL, Jiménez MA: Generalized Fibonacci Sequences and Their Matrix Representations. Linear Algebra Appl. 2022; 647.

[13] 13. Bernstein DJ: Fast Multiplication and Its Applications. Algorithmic Number Theory. 2008; 44: 325–384. Publisher Full Text

[14] 14. Horadam AF: Basic Properties of a Certain Generalized Sequence of Numbers. Fibonacci Q. 1965; 3: 161–176. Publisher Full Text

[15] 15. Benjamin AT, Quinn JJ: Proofs That Really Count: The Art of Combinatorial Proof. Mathematical Association of America; 2003.

[16] 16. Lemma M, Lambright J, Andreou S: The Mathematical Magic the Mulatu Numbers. Arts humanities hawaii. 2012; 1–11.

[17] 17. Lemma M, Lambright J: The fascinating mathematical identities relating the Mulatu and Fibonacci numbers. Adv. Appl. Math. Sci. 2012; 11(7): 285–290.

[18] 18. Galvez FJ, Dehesa JS: Novel Properties of Fibonacci and Lucas Polynomials. Math. Proc. Camb. Philos. Soc. 1985; 97: 159–164. Publisher Full Text

[19] 19. Filipponi P, Horadam AF: Second Derivative Sequences of Fibonacci and Lucas Polynomials. Fibonacci Q. 1993; 31(3): 194–204. Publisher Full Text

[20] 20. Costa EA, da Costa Mesquita ÉG , Catarino PMMC: On the connections between Fibonacci and Mulatu Numbers. Intermaths. 2025; 6(1): 17–36. Publisher Full Text

[21] 21. Stanley RP: Enumerative Combinatorics. Cambridge University Press; 2nd ed2011; 1. .

[22] 22. Lenstra HW, Shallit JO: Continued Fractions and Linear Recurrence Sequences. J Theor Nombres Bordeaux. 2022; 34(2).

[23] 23. Hoo KH, Madanshekaf AS: Applications of Fibonacci Sequences in Coding Theory. J. Discret. Math. Sci. Cryptogr. 2021; 24(2): 457–470.

Mulatu Polynomials and an Efficient Detection Algorithm for Mulatu Numbers

Abstract

Background

Methods

Results

Conclusions

Keywords

1. Introduction

1.1 Literature review

2. Preliminaries

3. Main results

3.1 Detection algorithms for special numbers

Theorem:

Proof:

Corollary:

Proof:

Proposition:

Lemma:

Theorem:

Proof:

Theorem:

Theorem:

Proof:

Theorem:

Theorem:

3. Conclusion and future work

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