On Generalized Hexanacci and Gaussian Generalized Hexanacci Numbers

Abstract

In this paper, we present Binet's formulas, generating functions, and the summation formulas for generalized Hexanacci numbers, and as special cases, we investigate Hexanacci and Hexanacci-Lucas numbers with their properties. Also, we define Gaussian generalized Hexanacci numbers and as special cases, we investigate Gaussian Hexanacci and Gaussian Hexanacci-Lucas numbers with their properties. Moreover, we give some identities for these numbers. Furthermore, we present matrix formulations of generalized Hexanacci numbers and Gaussian generalized Hexanacci numbers.

Keywords

• Hexanacci numbers
• Gaussian generalized Hexanacci numbers
• Gaussian Hexanacci numbers
• Gaussian Hexanacci-Lucas number

References

1. [1] Asci, M., Gurel E., Gaussian Jacobsthal and Gaussian Jacobsthal Polynomials, Notes on Number Theory and Discrete Mathematics, 19(2013), 25-36.
2. [2] Bacani, J. B., Rabago, J. F. T., On Generalized Fibonacci Numbers, Applied Mathematical Sciences,9(25)(2015), 3611-3622.
3. [3] Berzsenyi, G., Gaussian Fibonacci Numbers, Fibonacci Quarterly, 15(3)(1977), 233-236.
4. [4] Catarino, P., Campos, H., A note on Gaussian Modified Pell numbers, Journal of Information & Optimization Sciences, 39(6)(2018), 1363- 1371.
5. [5] Dresden, G. P., Du, Z., A Simplified Binet Formula for k-Generalized Fibonacci Numbers, Journal of Integer Sequences, 17(4)(2014), 1-9.
6. [6] Fraleigh, J. B., A First Course In Abstract Algebra, (2nd ed.), Addison-Wesley, Reading, ISBN 0-201-01984-1, 1976.
7. [7] Gurel, E., k-Order Gaussian Fibonacci and k-Order Gaussian Lucas Recurrence Relations, Ph.D Thesis, Pamukkale University Institute of Science Mathematics, Denizli, Turkey, 2015.
8. [8] Halici, S., Öz, S., On some Gaussian Pell and Pell-Lucas numbers, Ordu University Science and Technology Journal, 6(1)(2016), 8-18.

9. [9] Halici, S., Öz, S., On Gaussian Pell Polynomials and Their Some Properties, Palestine Journal of Mathematics, 7(1)(2018), 251-256.
10. [10] Harman, C. J., Complex Fibonacci Numbers, Fibonacci Quarterly, 19(1)(1981), 82-86.
11. [11] Horadam, A. F., Complex Fibonacci Numbers and Fibonacci quaternions, American Mathematical Monthly, 70(1963), 289-291.
12. [12] Jordan, J. H., Gaussian Fibonacci and Lucas Numbers, Fibonacci Quarterly, 3(1965), 315-318.
13. [13] Karaaslan, N., Ya˘gmur, T., Gaussian (s,t)-Pell and Pell-Lucas Sequences and Their Matrix Representations, BEU Journal of Science, 8(1)(2019), 46-59.
14. [14] Natividad, L. R., On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order, International Journal of Mathematics and Computing, 3(2)(2013), 38-40.
15. [15] Pethe S., Horadam A.F., Generalised Gaussian Fibonacci numbers, Bulletin of the Australian Mathematical Society, 33(1986), 37-48.
16. [16] Pethe S., Horadam, A. F., Generalised Gaussian Lucas Primordial numbers, Fibonacci Quarterly, (1988), 20-30. 1988.
17. [17] Pethe, S., Some Identities for Tribonacci Sequences, The Fibonacci Quarterly, 26(1988), 144-151.
18. [18] Rathore, G. P. S., Sikhwal, O., Choudhary, R., Formula for finding nth Term of Fibonacci-Like Sequence of Higher Order, International Journal of Mathematics And its Applications, 4(2-D)(2016), 75-80.
19. [19] Simson. R., An Explanation of an Obscure Passage in Albrecht Girard’s Commentary upon Simon Stevin’s Works, Philosophical Transactions of the Royal Society, 48(1)(1753), 368-377.
20. [20] Sloane, N. J. A., The on-line encyclopedia of integer sequences, http://oeis.org/.
21. [21] Soykan, Y., Taşdemir, E., Okumuş, İ., Göcen, M., Gaussian Generalized Tribonacci Numbers, Journal of Progressive Research in Mathematics, 14(2)(2018), 2373-2387.
22. [22] Soykan, Y., Gaussian Generalized Tetranacci Numbers, Journal of Advances in Mathematics and Computer Science, 31(3)(2019), 1-21.
23. [23] Soykan, Y., On Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers, Asian Research Journal of Mathematics, 16(9)(2020), 102-121.
24. [24] Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, International Journal of Advances in Applied Mathematics and Mechanics, 7(2)(2019), 45-56.
25. [25] Tas¸cı, D., Acar, H., Gaussian Tetranacci Numbers, Communications in Mathematics ans Applications, 8(3)(2017), 379-386.
26. [26] Tas¸cı, D., Acar, H., Gaussian Padovan and Gaussian Pell-Padovan Numbers, Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 67(2)(2018), 82-88.
27. [27] Yagmur, T., Karaaslan, N., Gaussian Modified Pell Sequence and Gaussian Modified Pell Polynomial Sequence, Aksaray University Journal of Science and Engineering, 2(1)(2018), 63-72.
28. [28] Wolfram, D. A., Solving Generalized Fibonacci Recurrences, Fibonacci Quarterly, (1998), 129-145.