Generalized Pierre Numbers
Keywords:
11B37, 11B39, 11B83
Abstract
In this paper, we introduce and investigate the generalized Pierre sequences for the first time and we deal with, in detail, two special cases, namely, Pierre and Pierre-Lucas 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. Furthermore, we show that there are close relations between Pierre, Pierre-Lucas and Tribonacci, Tribonacci-Lucas numbers.
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References
Bruce, I., A modified Tribonacci sequence, Fibonacci Quarterly, 22(3), 244--246, 1984.
Catalani, M., Identities for Tribonacci-related sequences, arXiv:math/0209179, 2012.
Choi, E., Modular Tribonacci Numbers by Matrix Method, Journal of the Korean Society of Mathematical Education Series B: Pure and Applied. Mathematics. 20(3), 207--221, 2013.
Elia, M., Derived Sequences, The Tribonacci Recurrence and Cubic Forms, Fibonacci Quarterly, 39 (2), 107-115, 2001.
Er, M. C., Sums of Fibonacci Numbers by Matrix Methods, Fibonacci Quarterly, 22(3), 204-207, 1984.
G. S. Hathiwala, D. V. Shah, Binet--Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods, Mathematical Journal of Interdisciplinary Sciences 6(1) (2017), 37--48.
Howard F.T., Saidak, F., Zhou's Theory of Constructing Identities, Congress Numer. 200 (2010), 225-237.
Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
Melham, R.S., Some Analogs of the Identity F_{n}²+F_{n+1}²=F_{2n+1}², Fibonacci Quarterly, (1999), 305-311.
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.
Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
Scott, A., Delaney, T., Hoggatt Jr., V., The Tribonacci sequence, Fibonacci Quarterly, 15(3), 193--200, 1977.
Shannon, A., Tribonacci numbers and Pascal's pyramid, Fibonacci Quarterly, 15(3), pp. 268 and 275, 1977.
B. Singh, P. Bhadouria, O. Sikhwal, K. Sisodiya, A Formula for Tetranacci-Like Sequence, Gen. Math. Notes, 20(2) (2014), 136-141.
Sloane, N.J.A., The on-line encyclopedia of integer sequences, http://oeis.org/
Soykan, Y. Tribonacci and Tribonacci-Lucas Sedenions. Mathematics 7(1), 74, 2019.
Soykan, Y., Summing Formulas For Generalized Tribonacci Numbers, Universal Journal of Mathematics and Applications, 3(1), 1-11, 2020. DOI: https://doi.org/10.32323/ujma.637876
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, International Journal of Advances in Applied Mathematics and Mechanics, 7(2), 45-56, 2019.
Soykan, Y., On Four Special Cases of Generalized Tribonacci Sequence: Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas Sequences, Journal of Progressive Research in Mathematics, 16(3), 3056-3084, 2020.
Soykan, Y., Properties of Generalized (r,s,t,u)-Numbers, Earthline Journal of Mathematical Sciences, 5(2), 297-327, 2021. https://doi.org/10.34198/ejms.5221.297327
Soykan, Y., A Study On the Recurrence Properties of Generalized Tetranacci Sequence, International Journal of Mathematics Trends and Technology, 67(8), 185-192, 2021. doi:10.14445/22315373/IJMTT-V67I8P522
Soykan, Y., A Study on Generalized Fibonacci Polynomials: Sum Formulas, International Journal of Advances in Applied Mathematics and Mechanics, 10(1), 39-118, 2022. (ISSN: 2347-2529)
Soykan, Y., On Generalized Fibonacci Polynomials: Horadam Polynomials, Earthline Journal of Mathematical Sciences, 11(1), 23-114, 2023. E-ISSN: 2581-8147. https://doi.org/10.34198/ejms.11123.23114
Soykan, Y. On the Recurrence Properties of Generalized Tribonacci Sequence, Earthline Journal of Mathematical Sciences, 6(2), 253-269, 2021. https://doi.org/10.34198/ejms.6221.253269
Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quarterly, 10(3), 231--246, 1972.
Yilmaz, N., Taskara, N., Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices, Applied Mathematical Sciences, 8(39), 1947-1955, 2014.
M. E. Waddill, Another Generalized Fibonacci Sequence, M. E., Fibonacci Quarterly, 5(3) (1967), 209-227.
M. E. Waddill, The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, (1992), 9-20.
Catalani, M., Identities for Tribonacci-related sequences, arXiv:math/0209179, 2012.
Choi, E., Modular Tribonacci Numbers by Matrix Method, Journal of the Korean Society of Mathematical Education Series B: Pure and Applied. Mathematics. 20(3), 207--221, 2013.
Elia, M., Derived Sequences, The Tribonacci Recurrence and Cubic Forms, Fibonacci Quarterly, 39 (2), 107-115, 2001.
Er, M. C., Sums of Fibonacci Numbers by Matrix Methods, Fibonacci Quarterly, 22(3), 204-207, 1984.
G. S. Hathiwala, D. V. Shah, Binet--Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods, Mathematical Journal of Interdisciplinary Sciences 6(1) (2017), 37--48.
Howard F.T., Saidak, F., Zhou's Theory of Constructing Identities, Congress Numer. 200 (2010), 225-237.
Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
Melham, R.S., Some Analogs of the Identity F_{n}²+F_{n+1}²=F_{2n+1}², Fibonacci Quarterly, (1999), 305-311.
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.
Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
Scott, A., Delaney, T., Hoggatt Jr., V., The Tribonacci sequence, Fibonacci Quarterly, 15(3), 193--200, 1977.
Shannon, A., Tribonacci numbers and Pascal's pyramid, Fibonacci Quarterly, 15(3), pp. 268 and 275, 1977.
B. Singh, P. Bhadouria, O. Sikhwal, K. Sisodiya, A Formula for Tetranacci-Like Sequence, Gen. Math. Notes, 20(2) (2014), 136-141.
Sloane, N.J.A., The on-line encyclopedia of integer sequences, http://oeis.org/
Soykan, Y. Tribonacci and Tribonacci-Lucas Sedenions. Mathematics 7(1), 74, 2019.
Soykan, Y., Summing Formulas For Generalized Tribonacci Numbers, Universal Journal of Mathematics and Applications, 3(1), 1-11, 2020. DOI: https://doi.org/10.32323/ujma.637876
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, International Journal of Advances in Applied Mathematics and Mechanics, 7(2), 45-56, 2019.
Soykan, Y., On Four Special Cases of Generalized Tribonacci Sequence: Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas and adjusted Tribonacci-Lucas Sequences, Journal of Progressive Research in Mathematics, 16(3), 3056-3084, 2020.
Soykan, Y., Properties of Generalized (r,s,t,u)-Numbers, Earthline Journal of Mathematical Sciences, 5(2), 297-327, 2021. https://doi.org/10.34198/ejms.5221.297327
Soykan, Y., A Study On the Recurrence Properties of Generalized Tetranacci Sequence, International Journal of Mathematics Trends and Technology, 67(8), 185-192, 2021. doi:10.14445/22315373/IJMTT-V67I8P522
Soykan, Y., A Study on Generalized Fibonacci Polynomials: Sum Formulas, International Journal of Advances in Applied Mathematics and Mechanics, 10(1), 39-118, 2022. (ISSN: 2347-2529)
Soykan, Y., On Generalized Fibonacci Polynomials: Horadam Polynomials, Earthline Journal of Mathematical Sciences, 11(1), 23-114, 2023. E-ISSN: 2581-8147. https://doi.org/10.34198/ejms.11123.23114
Soykan, Y. On the Recurrence Properties of Generalized Tribonacci Sequence, Earthline Journal of Mathematical Sciences, 6(2), 253-269, 2021. https://doi.org/10.34198/ejms.6221.253269
Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
Yalavigi, C. C., Properties of Tribonacci numbers, Fibonacci Quarterly, 10(3), 231--246, 1972.
Yilmaz, N., Taskara, N., Tribonacci and Tribonacci-Lucas Numbers via the Determinants of Special Matrices, Applied Mathematical Sciences, 8(39), 1947-1955, 2014.
M. E. Waddill, Another Generalized Fibonacci Sequence, M. E., Fibonacci Quarterly, 5(3) (1967), 209-227.
M. E. Waddill, The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, (1992), 9-20.
Published
2023-02-19
How to Cite
Soykan, Y. (2023). Generalized Pierre Numbers. Journal of Progressive Research in Mathematics, 20(1), 16-38. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2182
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