Gaussian Generalized Tribonacci Numbers

  • İnci Okumuş Department of Mathematics, Faculty of Art and Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak
  • Yüksel Soykan Department of Mathematics, Faculty of Art and Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak
  • Erkan Taşdemir Kırklareli University, Pınarhisar Vocational School of Higher Education, 39300, Kırklareli
  • Melih Göcen Department of Mathematics, Faculty of Art and Science, Zonguldak Bülent Ecevit University, 67100, Zonguldak
Keywords: Tribonacci numbers, Gaussian generalized Tribonacci numbers, Gaussian Tribonacci numbers, Gaussian Tribonacci-Lucas numbers.


In this paper, we define Gaussian generalized Tribonacci numbers and as special cases, we investigate Gaussian Tribonacci and Gaussian Tribonacci-Lucas numbers with their properties.


[1] Asci, M., and Gurel, E., Gaussian Jacobsthal and Gaussian Jacobsthal Polynomials, Notes on Number Theory and Discrete Mathematics, Vol.19, pp.25-36, 2013.
[2] Basu, M., Das, M., Tribonacci Matrices and a New Coding Theory, Discrete Mathematics, Algorithms and Applications, Vol. 6, No. 1, 1450008, (17 pages), 2014.
[3] Berzsenyi, G., Gaussian Fibonacci Numbers, Fibonacci Quart., Vol.15(3), pp.233-236, 1977.
[4] Bruce, I., A modified Tribonacci sequence, The Fibonacci Quarterly, 22 : 3, pp. 244-246, 1984.
[5] Catalani, M., Identities for Tribonacci-related sequences - arXiv preprint,, 2002.
[6] Catarino, P., and Campos, H., A note on Gaussian Modified Pell numbers, Journal of Information & Optimization Sciences, Vol. 39, No. 6, pp. 1363-1371, 2018.
[7] Choi, E., Modular tribonacci Numbers by Matrix Method, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. Volume 20, Number 3 (August 2013), Pages 207-221, 2013.
[8] Elia, M., Derived Sequences, The Tribonacci Recurrence and Cubic Forms, The Fibonacci Quarterly, 39:2, pp. 107-115, 2001.
[9] Falcon, S., On the complex k-Fibonacci numbers, Cogent Mathematics, 3: 1201944, 9 pages, 2016.
[10] Fraleigh, J.B., A First Course In Abstract Algebra, (2nd ed.), Addison-Wesley, Reading, ISBN 0-201-01984-1, 1976.
[11] Frontczak, R., Convolutions for Generalized Tribonacci Numbers and Related Results, International Journal of Mathematical Analysis, Vol. 12, 2018, no. 7, 307-324.
[12] Gurel, E., k-Order Gaussian Fibonacci and k-Order Gaussian Lucas Recurrence Relations, PhD Thesis, Pamukkale University Institute of Science Mathematics, Denizli, Turkey (2015).
[13] Halici, S., Öz, S., On Some Gaussian Pell and Pell-Lucas Numbers, Ordu University Science and Technology Journal, Vol.6(1), pp.8-18, 2016.
[14] Halici, S., Öz, S., On Gaussian Pell Polynomials and Their Some Properties, Palastine Journal of Mathematics, Vol 7(1), 251-256, 2018.
[15] Harman, C.J., Complex Fibonacci Numbers, Fibonacci Quart., Vol.19(1), pp. 82-86, 1981.
[16] Horadam, A.F., Complex Fibonacci Numbers and Fibonacci quaternions, Amer. Math. Monthly 70, 289-291, 1963.
[17] Jordan, J.H., Gaussian Fibonacci and Lucas Numbers, Fibonacci Quart., Vol.3, pp. 315-318, 1965.
[18] Lin, P.Y., De Moivre-Type Identities For The Tribonacci Numbers, The Fibonacci Quarterly, 26, pp. 131-134, 1988.
[19] Pethe, S., Horadam, A.F., Generalised Gaussian Fibonacci numbers, Bull. Austral. Math. Soc., Vol.33, pp.37-48, 1986.
[20] Pethe, S., Some Identities, The Fibonacci Quarterly, 26, pp. 144-246, 1988.
[21] Pethe, S., Horadam, A.F., Generalised Gaussian Lucas Primordial numbers, Fibonacci Quart., pp. 20-30, 1988.
[22] Pethe, S., Some Identities for Tribonacci Sequences, The Fibonacci Quarterly, 26, 144-151, 1988.
[23] Scott, A., Delaney, T., Hoggatt Jr., V., The Tribonacci sequence, The Fibonacci Quarterly, 15:3, pp. 193-200, 1977.
[24] Shannon, A.G, Horadam, A.F., Some Properties of Third-Order Recurrence Relations, The Fibonacci Quarterly, 10(2),, pp. 135-146, 1972.
[25] Shannon, A., Tribonacci numbers and Pascal's pyramid, The Fibonacci Quarterly, 15:3, pp. 268-275, 1977.
[26] Sloane, N.J.A., The on-line encyclopedia of integer sequences, arXiv preprint-1805.10343, 2018.
[27] Spickerman, W., Binet's formula for the Tribonacci sequence, The Fibonacci Quarterly, 20, pp.118-120, 1981.
[28] Taşcı, D., Acar, H., Gaussian Tetranacci Numbers, Communications in Mathematics ans Applications, Vol. 8, No. 3, pp. 379-386, 2017.
[29] Taşcı, D., Acar, H., Gaussian Padovan and Gaussian Pell-Padovan Numbers, Commun. Fac. Sci. Ank. Ser. A1 Math. Stat., Volume 67, Number 2, pp. 82-88, 2018.
[30] Yagmur, T., Karaaslan, N., Aksaray University Journal of Science and Engineering, Volume 2, Issue 1, pp. 63-72, 2018.
[31] Yalavigi, C.C., A Note on `Another Generalized Fibonacci Sequence', The Mathematics Student. 39, 407-408, 1971.
[32] Yalavigi, C.C., Properties of Tribonacci numbers, The Fibonacci Quarterly, 10 : 3, pp. 231--246, 1972.
[33] Yilmaz, N., Taskara, N., Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices, Applied Mathematical Sciences, 8, no. 39, 1947-1955, 2014.
[34] Waddill, M.E., Using Matrix Techniques to Establish Properties of a Generalized Tribonacci Sequence (in Applications of Fibonacci Numbers, Volume 4, G. E. Bergum et al., eds.). Kluwer Academic Publishers. Dordrecht, The Netherlands: pp. 299-308, 1991.
How to Cite
Okumuş, İnci, Soykan, Y., Taşdemir, E., & Göcen, M. (2018). Gaussian Generalized Tribonacci Numbers. Journal of Progressive Research in Mathematics, 14(2), 2373-2387. Retrieved from