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Journal articles on the topic 'Tribonacci numbers'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Yüksel, Soykan, Taşdemir Erkan, Okumuş İnci, and Göcen Melih. "Gaussian Generalized Tribonacci Numbers." Journal of Progressive Research in Mathematics 14, no. 2 (2018): 2373–87. https://doi.org/10.5281/zenodo.3974192.

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2

Tasyurdu, Yasemin. "Tribonacci and Tribonacci-Lucas hybrid numbers." International Journal of Contemporary Mathematical Sciences 14, no. 4 (2019): 245–54. http://dx.doi.org/10.12988/ijcms.2019.91124.

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3

Kızılaslan, Gonca, and Leyla Karabulut. "Unrestricted Tribonacci and Tribonacci–Lucas quaternions." Notes on Number Theory and Discrete Mathematics 29, no. 2 (2023): 310–21. http://dx.doi.org/10.7546/nntdm.2023.29.2.310-321.

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We define a generalization of Tribonacci and Tribonacci–Lucas quaternions with arbitrary Tribonacci numbers and Tribonacci–Lucas numbers coefficients, respectively. We get generating functions and Binet’s formulas for these quaternions. Furthermore, several sum formulas and a matrix representation are obtained.

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4

Adegoke, Kunle, Adenike Olatinwo, and Winning Oyekanmi. "New Tribonacci recurrence relations and addition formulas." Notes on Number Theory and Discrete Mathematics 26, no. 4 (2020): 164–72. http://dx.doi.org/10.7546/nntdm.2020.26.4.164-172.

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Only one three-term recurrence relation, namely, W_{r}=2W_{r-1}-W_{r-4}, is known for the generalized Tribonacci numbers, W_r, r\in Z, defined by W_{r}=W_{r-1}+W_{r-2}+W_{r-3} and W_{-r}=W_{-r+3}-W_{-r+2}-W_{-r+1}, where W_0, W_1 and W_2 are given, arbitrary integers, not all zero. Also, only one four-term addition formula is known for these numbers, which is W_{r + s} = T_{s - 1} W_{r - 1} + (T_{s - 1} + T_{s-2} )W_r + T_s W_{r + 1}, where ({T_r})_{r\in Z} is the Tribonacci sequence, a special case of the generalized Tribonacci sequence, with W_0 = T_0 = 0 and W_1 = W_2 = T_1 = T_2 = 1. In th

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5

Frontczak, Robert. "Sums of Tribonacci and Tribonacci-Lucas numbers." International Journal of Mathematical Analysis 12, no. 1 (2018): 19–24. http://dx.doi.org/10.12988/ijma.2018.712153.

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6

Erduvan, Fatih. "Tribonacci numbers as sum or difference of powers of 2." Celal Bayar Üniversitesi Fen Bilimleri Dergisi 21, no. 2 (2025): 147–51. https://doi.org/10.18466/cbayarfbe.1528991.

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This paper investigates Tribonacci numbers can be expressed as either the sum or difference of two distinct powers of 2. Namely, we address the problem of expressing Tribonacci numbers in the form T_n=2^x±2^y in positive integers with 1≤y≤x. Our findings reveal specific instances where such representations are possible, including examples like the seventh Tribonacci number expressed both as the sum and the difference of powers of 2. Additionally, we identify Tribonacci numbers that can be represented as the differences of Mersenne numbers, specifically, the numbers 2, 4, 24, and 504. These res

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7

Demirci, Musa, and Ismail Naci Cangul. "Tribonacci graphs." ITM Web of Conferences 34 (2020): 01002. http://dx.doi.org/10.1051/itmconf/20203401002.

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Special numbers have very important mathematical properties alongside their numerous applications in many ﬁelds of science. Probably the most important of those is the Fibonacci numbers. In this paper, we use a generalization of Fibonacci numbers called tribonacci numbers having very limited properties and relations compared to Fibonacci numbers. There is almost no result on the connections between these numbers and graphs. A graph having a degree sequence consisting of t successive tribonacci numbers is called a tribonacci graph of order t. Recently, a new graph parameter named as omega invar

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8

Yüksel, Soykan. "On Four Special Cases of Generalized Tribonacci Sequence: Tribonacci-Perrin, modified Tribonacci, modified TribonacciLucas and adjusted Tribonacci-Lucas Sequences." Journal of Progressive Research in Mathematics 16, no. 3 (2020): 3056–84. https://doi.org/10.5281/zenodo.3973345.

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In this paper, we investigate four new special cases, namely, Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas, adjusted Tribonacci-Lucas sequences, of the generalized Tribonacci sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

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9

Hulku, Sakıne, and Ömür Devec. "The Tribonacci-type balancing numbers and their applications." Mathematica Moravica 27, no. 1 (2023): 23–36. http://dx.doi.org/10.5937/matmor2301023h.

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N this paper, we define the Tribonacci-type balancing numbers via a Diophantine equation with a complex variable and then give their miscellaneous properties. Also, we study the Tribonacci-type balancing sequence modulo m and then obtain some interesting results concerning the periods of the Tribonacci-type balancing sequences for any m. Furthermore, we produce the cyclic groups using the multiplicative orders of the generating matrices of the Tribonacci-type balancing numbers when read modulo m. Then give the connections between the periods of the Tribonacci-type balancing sequences modulo m

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10

BASU, MANJUSRI, and MONOJIT DAS. "TRIBONACCI MATRICES AND A NEW CODING THEORY." Discrete Mathematics, Algorithms and Applications 06, no. 01 (2014): 1450008. http://dx.doi.org/10.1142/s1793830914500086.

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In this paper, we consider the series of Tribonacci numbers. Thereby, we introduce a new coding theory called Tribonacci coding theory based on Tribonacci numbers and show that in the simplest case, the correct ability of this method is 99.80% whereas the correct ability of the Fibonacci coding/decoding method is 93.33%.

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11

Badidja, Salim, Ahmed Ait Mokhtar, and Özen Özer. "Representation of integers by k-generalized Fibonacci sequences and applications in cryptography." Asian-European Journal of Mathematics 14, no. 09 (2021): 2150157. http://dx.doi.org/10.1142/s1793557121501576.

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The aim of this paper is to construct a relation between tribonacci numbers and generalized tribonacci numbers. Besides, certain conditions are obtained to generalize the representation of a positive integer [Formula: see text] which is determined in [S. Badidja and A. Boudaoud, Representation of positive integers as a sum of distinct tribonacci numbers, J. Math. Statistic. 13 (2017) 57–61] for a [Formula: see text]-generalized Fibonacci numbers [Formula: see text]. Lastly, some applications to cryptography are given by using [Formula: see text].

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12

Choi, Eunmi, and Jiin Jo. "IDENTITIES INVOLVING TRIBONACCI NUMBERS." Journal of the Chungcheong Mathematical Society 28, no. 1 (2015): 39–51. http://dx.doi.org/10.14403/jcms.2015.28.1.39.

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13

Yüksel, Soykan. "A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers." Journal of Progressive Research in Mathematics 16, no. 2 (2020): 2932–41. https://doi.org/10.5281/zenodo.3973991.

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In this paper, closed forms of the sum formulas for the squares of generalized Tribonacci numbers are presented. As special cases, we give summation formulas of the squares of Tribonacci, Tribonacci Lucas, Padovan, Perrin, Narayana and some other third order linear recurrence sequences.

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14

Choi, Eunmi, and Jiin Jo. "On Partial Sum of Tribonacci Numbers." International Journal of Mathematics and Mathematical Sciences 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/301814.

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We study the sumst(k,r)=∑i=0tTki+rofkstep apart Tribonacci numbers for any1≤r≤k. We prove thatst(k,r)satisfies certain Tribonacci rulest(k,r)=akst-1(k,r)+bkst-2(k,r)+st-3(k,r)+λwith integersak,bk,ck, andλ.

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15

Tasyurdu, Yasemin. "On the sums of Tribonacci and Tribonacci-Lucas numbers." Applied Mathematical Sciences 13, no. 24 (2019): 1201–8. http://dx.doi.org/10.12988/ams.2019.910144.

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16

Zhou, Yuetong, Peng Yang, Shaonan Zhang, and Kaiqiang Zhang. "Repdigits as Sums of Four Tribonacci Numbers." Symmetry 14, no. 9 (2022): 1931. http://dx.doi.org/10.3390/sym14091931.

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In this paper, we show that 66666 is the largest repdigit expressible as the sum of four tribonacci numbers. We used Binet’s formula, Baker’s theory, and a reduction method during the proving procedure. We also used the periodic properties of tribonacci number modulo 9 to deal with three individual cases.

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17

Chelgham, Mourad, Ali Boussayoud, and Kasi Viswanadh V. Kanuri. "A new class of symmetric functions of binary products of tribonacci numbers and other well-known numbers." Online Journal of Analytic Combinatorics, no. 16 (December 31, 2021): 1–17. https://doi.org/10.61091/ojac-1607.

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In this paper, we will recover the generating functions of Tribonacci numbers and Chebychev polynomials of first and second kind. By making use of the operator defined in this paper, we give some new generating functions for the binary products of Tribonacci with some remarkable numbers and polynomials. The technique used here is based on the theory of the so-called symmetric functions.

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18

Li, Juan, Zhaolin Jiang, and Fuliang Lu. "Determinants, Norms, and the Spread of Circulant Matrices with Tribonacci and Generalized Lucas Numbers." Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/381829.

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Circulant matrices play an important role in solving ordinary and partial differential equations. In this paper, by using the inverse factorization of polynomial of degreen, the explicit determinants of circulant and left circulant matrix involving Tribonacci numbers or generalized Lucas numbers are expressed in terms of Tribonacci numbers and generalized Lucas numbers only. Furthermore, four kinds of norms and bounds for the spread of these matrices are given, respectively.

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19

Bravo, Eric F., and Jhon J. Bravo. "Tribonacci numbers with two blocks of repdigits." Mathematica Slovaca 71, no. 2 (2021): 267–74. http://dx.doi.org/10.1515/ms-2017-0466.

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Abstract The Tribonacci sequence is a generalization of the Fibonacci sequence which starts with 0,0,1 and each term afterwards is the sum of the three preceding terms. Here, we show that the only Tribonacci numbers that are concatenations of two repdigits are 13,24,44,81. This paper continues a previous work that searched for Fibonacci numbers which are concatenations of two repdigits.

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20

Yilmaz, Nazmiye, and Necati Taskara. "Tribonacci and Tribonacci-Lucas numbers via the determinants of special matrices." Applied Mathematical Sciences 8 (2014): 1947–55. http://dx.doi.org/10.12988/ams.2014.4270.

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21

Yüksel, Soykan, and Okumuş İnci. "On a Generalized Tribonacci Sequence." Journal of Progressive Research in Mathematics 14, no. 3 (2019): 2413–18. https://doi.org/10.5281/zenodo.3974099.

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The well-known Tribonacci sequence is a third order recurrence sequence. In this paper, we define other generalized Tribonacci sequence and establish some properties of this sequence using matrix methods.

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22

Yüksel Soykan, Melih Göcen, and İnci Okumuş. "On Tribonacci functions and Gaussian Tribonacci functions." Malaya Journal of Matematik 11, S (2023): 208–26. http://dx.doi.org/10.26637/mjm11s/013.

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In this work, Gaussian Tribonacci functions are defined and investigated on the set of real numbers $\mathbb{R}$, i.e., functions $f_G: \mathbb{R} \rightarrow \mathbb{C}$ such that for all $x \in \mathbb{R}, n \in \mathbb{Z}, f_G(x+n)=$ $f(x+n)+i f(x+n-1)$ where $f: \mathbb{R} \rightarrow \mathbb{R}$ is a Tribonacci function which is given as $f(x+3)=$ $f(x+2)+f(x+1)+f(x)$ for all $x \in \mathbb{R}$. Then the concept of Gaussian Tribonacci functions by using the concept of $f$-even and $f$-odd functions is developed. Also, we present linear sum formulas of Gaussian Tribonacci functions. Moreov

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23

Choi, Eunmi. "MODULAR TRIBONACCI NUMBERS BY MATRIX METHOD." Pure and Applied Mathematics 20, no. 3 (2013): 207–21. http://dx.doi.org/10.7468/jksmeb.2013.20.3.207.

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24

Cereceda, José Luis. "Binet's formula for generalized tribonacci numbers." International Journal of Mathematical Education in Science and Technology 46, no. 8 (2015): 1235–43. http://dx.doi.org/10.1080/0020739x.2015.1031837.

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25

Gómez Ruiz, Carlos Alexis, and Florian Luca. "Multiplicatively dependent triples of Tribonacci numbers." Acta Arithmetica 171, no. 4 (2015): 327–53. http://dx.doi.org/10.4064/aa171-4-3.

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26

Yilmaz, Nazmiye, and Necati Taskara. "Incomplete Tribonacci–Lucas Numbers and Polynomials." Advances in Applied Clifford Algebras 25, no. 3 (2014): 741–53. http://dx.doi.org/10.1007/s00006-014-0523-8.

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27

Sivaraman Nair, Prabha, and Rejikumar Karunakaran. "On Brousseau Sums of Tribonacci Numbers." Fibonacci Quarterly 63, no. 1 (2025): 22–38. https://doi.org/10.1080/00150517.2024.2412961.

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28

Luca, Florian, Japhet Odjoumani, and Alain Togbé. "Tribonacci Numbers That Are Products of Two Fibonacci Numbers." Fibonacci Quarterly 61, no. 4 (2023): 298–304. http://dx.doi.org/10.1080/00150517.2023.12427384.

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29

Zhou, Shujie, and Li Chen. "Tribonacci Numbers and Some Related Interesting Identities." Symmetry 11, no. 10 (2019): 1195. http://dx.doi.org/10.3390/sym11101195.

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The main purpose of this paper is, by using elementary methods and symmetry properties of the summation procedures, to study the computational problem of a certain power series related to the Tribonacci numbers, and to give some interesting identities for these numbers.

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30

Evink, Tim, and Paul Alexander Helminck. "Tribonacci numbers and primes of the form p = x2 + 11y2." Mathematica Slovaca 69, no. 3 (2019): 521–32. http://dx.doi.org/10.1515/ms-2017-0244.

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Abstract In this paper we show that for any prime number p not equal to 11 or 19, the Tribonacci number Tp−1 is divisible by p if and only if p is of the form x2 + 11y2. We first use class field theory on the Galois closure of the number field corresponding to the polynomial x3 − x2 − x − 1 to give the splitting behavior of primes in this number field. After that, we apply these results to the explicit exponential formula for Tp−1. We also give a connection between the Tribonacci numbers and the Fourier coefficients of the unique newform of weight 2 and level 11.

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31

Yagmur, T\"{u}lay. "A note on generalized hybrid Tribonacci numbers." Discussiones Mathematicae - General Algebra and Applications 40, no. 2 (2020): 187. http://dx.doi.org/10.7151/dmgaa.1343.

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32

Shah, Devbhadra V., and Darshana A. Mehta. "Golden proportions for the generalized Tribonacci numbers." International Journal of Mathematical Education in Science and Technology 40, no. 6 (2009): 837–42. http://dx.doi.org/10.1080/00207390902971940.

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33

Šiurys, Jonas. "A Tribonacci-Like Sequence of Composite Numbers." Fibonacci Quarterly 49, no. 4 (2011): 298–302. http://dx.doi.org/10.1080/00150517.2011.12428028.

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34

Robbins, Neville. "On Tribonacci Numbers and 3-Regular Compositions." Fibonacci Quarterly 52, no. 1 (2014): 16–19. http://dx.doi.org/10.1080/00150517.2014.12427915.

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35

Choi, Eunmi. "A SEQUENCE OF SUMS OF TRIBONACCI NUMBERS." Far East Journal of Mathematical Sciences (FJMS) 109, no. 1 (2018): 37–56. http://dx.doi.org/10.17654/ms109010037.

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36

Mansour, Toufik, and Mark Shattuck. "Polynomials whose coefficients are generalized Tribonacci numbers." Applied Mathematics and Computation 219, no. 15 (2013): 8366–74. http://dx.doi.org/10.1016/j.amc.2012.12.052.

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37

Kanovich, Max. "Multiset rewriting over Fibonacci and Tribonacci numbers." Journal of Computer and System Sciences 80, no. 6 (2014): 1138–51. http://dx.doi.org/10.1016/j.jcss.2014.04.006.

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38

Došlić, Tomislav, and Luka Podrug. "Tilings of a honeycomb strip and higher order Fibonacci numbers." Contributions to Discrete Mathematics 19, no. 2 (2024): 56–81. http://dx.doi.org/10.55016/ojs/cdm.v19i2.75062.

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In this paper we explore two types of tilings of a honeycomb strip and derive some closed form formulas for the number of tilings. Furthermore, we obtain some new identities involving tribonacci numbers, Padovan numbers and Narayana's cow sequence and provide combinatorial proofs for several known identities about those numbers.

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39

Trojovský, Pavel. "On Repdigits as Sums of Fibonacci and Tribonacci Numbers." Symmetry 12, no. 11 (2020): 1774. http://dx.doi.org/10.3390/sym12111774.

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In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a ”symmetrical” type of numbers) that can be written in the form Fn+Tn, for some n≥1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively.

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40

Bednařík, Dušan, and Eva Trojovská. "Repdigits as Product of Fibonacci and Tribonacci Numbers." Mathematics 8, no. 10 (2020): 1720. http://dx.doi.org/10.3390/math8101720.

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In this paper, we study the problem of the explicit intersection of two sequences. More specifically, we find all repdigits (i.e., numbers with only one repeated digit in its decimal expansion) which can be written as the product of a Fibonacci by a Tribonacci number (both with the same indexes). To work on this problem, our approach is to combine lower bounds from the Baker’s theory with reduction methods (based on the theory of continued fractions) due to Dujella and Pethö.

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41

Gürses, Nurten, and Zehra İşbilir. "An extended framework for bihyperbolic generalized Tribonacci numbers." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 3 (2024): 765–86. http://dx.doi.org/10.31801/cfsuasmas.1378136.

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The aim of this article is to identify and analyze a new type special number system which is called bihyperbolic generalized Tribonacci numbers (BGTN for short). For this purpose, we give both classical and several new properties such as; recurrence relation, Binet formula, generating function, exponential generating function, summation formulae, matrix formula, and special determinant equations of BGTN . Also, the system of BGTN is quite a big family and includes several type special cases with respect to initial values and $r,~ s, ~t$ values, we give the subfamilies and special cases of it.

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42

Lin, Pin-Yen. "De Moivre-Type Identities for the Tribonacci Numbers." Fibonacci Quarterly 26, no. 2 (1988): 131–34. http://dx.doi.org/10.1080/00150517.1988.12429641.

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43

Cereceda, Jose Luis. "Determinantal representations for generalized Fibonacci and tribonacci numbers." International Journal of Contemporary Mathematical Sciences 9 (2014): 269–85. http://dx.doi.org/10.12988/ijcms.2014.4323.

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44

Frontczak, Robert. "Convolutions for generalized Tribonacci numbers and related results." International Journal of Mathematical Analysis 12, no. 7 (2018): 307–24. http://dx.doi.org/10.12988/ijma.2018.8429.

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45

Taher, Hunar, and Saroj Kumar Dash. "Common Terms of k-Pell and Tribonacci Numbers." European Journal of Pure and Applied Mathematics 17, no. 1 (2024): 135–46. http://dx.doi.org/10.29020/nybg.ejpam.v17i1.4989.

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Let Tm be a Tribonacci sequence, and let the k-Pell sequence be a generalization of the Pell sequence for k ≥ 2 . The first k terms are 0, 0, ..., 0, 1, and each term after the forewords is defined by linear recurrence P (k) n = 2P (k) n−1 + P (k) n−2 + ... + P (k) n−k . We study the solution of the Diophantine equation P (k) n = Tm for the positive integer (n, k, m) with k ≥ 2. We use the lower bound for linear forms in logarithms of algebraic numbers with the theory of the continued fraction.

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46

MERZOUK, HIND, ALI BOUSSAYOUD, and MOURAD CHELGHAM. "SYMMETRIC FUNCTIONS OF BINARY PRODUCTS OF TRIBONACCI LUCAS NUMBERS AND ORTHOGONAL POLYNOMIALS." Journal of Science and Arts 21, no. 2 (2021): 461–78. http://dx.doi.org/10.46939/j.sci.arts-21.2-a13.

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In this paper, we will recover the new generating functions of some products of Tribonacci Lucas numbers and orthogonal polynomials. The technic used her is based on the theory of the so called symmetric functions.

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47

Belovas, Igoris. "Central and Local Limit Theorems for Numbers of the Tribonacci Triangle." Mathematics 9, no. 8 (2021): 880. http://dx.doi.org/10.3390/math9080880.

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In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

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48

G\'OMEZ RUIZ, CARLOS ALEXIS, and FLORIAN LUCA. "Diophantine quadruples in the sequence of shifted Tribonacci numbers." Publicationes Mathematicae Debrecen 86, no. 3-4 (2015): 473–91. http://dx.doi.org/10.5486/pmd.2015.7118.

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49

Edwards, Kenneth. "A Pascal-Like Triangle Related to the Tribonacci Numbers." Fibonacci Quarterly 46-47, no. 1 (2008): 18–25. http://dx.doi.org/10.1080/00150517.2008.12428183.

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50

Irmak, Nurettin, and László Szalay. "Tribonacci Numbers Close to the Sum $2^a+3^b+5^c$." MATHEMATICA SCANDINAVICA 118, no. 1 (2016): 27. http://dx.doi.org/10.7146/math.scand.a-23293.

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We show that there are exactly 22 solutions to the inequalities \[ 0\le T_n-2^a-3^b-5^c\le10, \] where $T_n$ denotes the $n^{\it th}$ term ($n\ge0$) of the Tribonacci sequence, and $0\le a,b\le c$ are integers. All the solutions are explicitly determined.

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