A Study on Sum Formulas of Generalized Pentanacci Sequence: Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k}
Keywords:
Pentanacci numbers, Pentanacci-Lucas numbers, sum formulas, summing formulas.
Abstract
In this paper, closed forms of the sum formulas $\sum_{k=0}^{n}x^{k}W_{k}$ and $\sum_{k=1}^{n}x^{k}W_{-k}$ for generalized Pentanacci numbers are presented. As special cases, we give summation formulas of Pentanacci, Pentanacci-Lucas, and other fifth-order recurrence sequences.
Downloads
Download data is not yet available.
References
Akbulak, M., Öteleş. A., On the sum of Pell and Jacobsthal numbers by matrix method, Bulletin of the Iranian Mathematical Society, 40 (4), 1017-1025, 2014.
Cook C. K., Bacon, M. R., Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41, 27--39, 2013.
Frontczak, R., Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, 12 (1), 19-24, 2018.
Gökbaş, H., Köse, H., Some Sum Formulas for Products of Pell and Pell-Lucas Numbers, Int. J. Adv. Appl. Math. and Mech. 4(4), 1-4, 2017.
Hansen., R.T., General Identities for Linear Fibonacci and Lucas Summations, Fibonacci Quarterly, 16(2), 121-28, 1978.
Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, New York, 2001.
Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
Melham, R. S., Some Analogs of the Identity F_{n}²+F_{n+1}²=F_{2n+1}², Fibonacci Quarterly, 305-311, 1999.
Natividad, L. R., On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order, International Journal of Mathematics and Computing, 3 (2), 2013.
Parpar, T., k'ncı Mertebeden Rekürans Bağıntısının Özellikleri ve Bazı Uygulamaları, Selçuk Üniversitesi, Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 2011.
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), 75-80, 2016.
Sloane, N.J.A., The on-line encyclopedia of integer sequences. Available: http://oeis.org/
Soykan, Y., On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 20(2), 1-15, 2019.
Soykan, Y., Corrigendum: On Summing Formulas for Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 21(10), 66-82, 2020. DOI: 10.9734/AIR/2020/v21i1030253
Soykan,Y., On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports 8(1): 45-61, 2020, DOI: 10.9734/AJARR/2020/v8i130192.
Soykan, Y., Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science, 35(1), 89-104, 2020, DOI: 10.9734/JAMCS/2020/v35i130241.
Soykan Y., Generalized Tribonacci Numbers: Summing Formulas, Int. J. Adv. Appl. Math. and Mech. 7(3), 57-76, 2020.
Soykan, Y., Summing Formulas For Generalized Tribonacci Numbers, Universal Journal of Mathematics and Applications, 3(1), 1-11, 2020. ISSN 2619-9653, DOI: https://doi.org/10.32323/ujma.637876
Soykan, Y., On Sum Formulas for Generalized Tribonacci Sequence, Journal of Scientific Research & Reports, 26(7), 27-52, 2020. ISSN: 2320-0227, DOI: 10.9734/JSRR/2020/v26i730283
Soykan, Y., Summation Formulas For Generalized Tetranacci Numbers, Asian Journal of Advanced Research and Reports, 7(2), 1-12, 2019. doi.org/10.9734/ajarr/2019/v7i230170.
Soykan, Y., Sum Formulas For Generalized Fifth-Order Linear Recurrence Sequences, Journal of Advances in Mathematics and Computer Science, 34(5), 1-14, 2019; Article no.JAMCS.53303, ISSN: 2456-9968, DOI: 10.9734/JAMCS/2019/v34i530224.
Soykan, Y., Linear Summing Formulas of Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers, Journal of Advanced in Mathematics and Computer Science, 33(3): 1-14, 2019.
Soykan, Y., On Summing Formulas of Generalized Hexanacci and Gaussian Generalized Hexanacci Numbers, Asian Research Journal of Mathematics, 14(4), 1-14, 2019; Article no.ARJOM.50727.
Soykan, Y., A Study On Sum Formulas of Generalized Sixth-Order Linear Recurrence Sequences, Asian Journal of Advanced Research and Reports, 14(2), 36-48, 2020. DOI: 10.9734/AJARR/2020/v14i230329
Soykan, Y., Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, Communications in Mathematics and Applications, 11(2), 281-295, 2020. DOI: 10.26713/cma.v11i2.1102
Soykan, Y., On Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers, Asian Research Journal of Mathematics, 16(9), 102-121, 2020. DOI: 10.9734/ARJOM/2020/v16i930224
Soykan, Y., Properties of Generalized Fifth-Order Pell Numbers, Asian Research Journal of Mathematics, 15(3), 1-18, 2019.
Soykan, Y., Polatlı, E.E., A Note on Fifth Order Jacobsthal Numbers, IOSR Journal of Mathematics (IOSR-JM), 17(2), 01-23, 2021. DOI: 10.9790/5728-1702010123
Soykan, Y., A Study On Generalized 5-primes Numbers, Journal of Scientific Perspectives, 4(3), 185-202, 2020,, DOI: https://doi.org/10.26900/jsp.4.017.
Öteleş, A., Akbulak, M., A Note on Generalized k-Pell Numbers and Their Determinantal Representation, Journal of Analysis and Number Theory, 4(2), 153-158, 2016.
Waddill, M. E., The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, 9-20, 1992.
Cook C. K., Bacon, M. R., Some identities for Jacobsthal and Jacobsthal-Lucas numbers satisfying higher order recurrence relations, Annales Mathematicae et Informaticae, 41, 27--39, 2013.
Frontczak, R., Sums of Tribonacci and Tribonacci-Lucas Numbers, International Journal of Mathematical Analysis, 12 (1), 19-24, 2018.
Gökbaş, H., Köse, H., Some Sum Formulas for Products of Pell and Pell-Lucas Numbers, Int. J. Adv. Appl. Math. and Mech. 4(4), 1-4, 2017.
Hansen., R.T., General Identities for Linear Fibonacci and Lucas Summations, Fibonacci Quarterly, 16(2), 121-28, 1978.
Koshy, T., Fibonacci and Lucas Numbers with Applications, A Wiley-Interscience Publication, New York, 2001.
Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
Melham, R. S., Some Analogs of the Identity F_{n}²+F_{n+1}²=F_{2n+1}², Fibonacci Quarterly, 305-311, 1999.
Natividad, L. R., On Solving Fibonacci-Like Sequences of Fourth, Fifth and Sixth Order, International Journal of Mathematics and Computing, 3 (2), 2013.
Parpar, T., k'ncı Mertebeden Rekürans Bağıntısının Özellikleri ve Bazı Uygulamaları, Selçuk Üniversitesi, Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 2011.
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), 75-80, 2016.
Sloane, N.J.A., The on-line encyclopedia of integer sequences. Available: http://oeis.org/
Soykan, Y., On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 20(2), 1-15, 2019.
Soykan, Y., Corrigendum: On Summing Formulas for Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 21(10), 66-82, 2020. DOI: 10.9734/AIR/2020/v21i1030253
Soykan,Y., On Summing Formulas for Horadam Numbers, Asian Journal of Advanced Research and Reports 8(1): 45-61, 2020, DOI: 10.9734/AJARR/2020/v8i130192.
Soykan, Y., Generalized Fibonacci Numbers: Sum Formulas, Journal of Advances in Mathematics and Computer Science, 35(1), 89-104, 2020, DOI: 10.9734/JAMCS/2020/v35i130241.
Soykan Y., Generalized Tribonacci Numbers: Summing Formulas, Int. J. Adv. Appl. Math. and Mech. 7(3), 57-76, 2020.
Soykan, Y., Summing Formulas For Generalized Tribonacci Numbers, Universal Journal of Mathematics and Applications, 3(1), 1-11, 2020. ISSN 2619-9653, DOI: https://doi.org/10.32323/ujma.637876
Soykan, Y., On Sum Formulas for Generalized Tribonacci Sequence, Journal of Scientific Research & Reports, 26(7), 27-52, 2020. ISSN: 2320-0227, DOI: 10.9734/JSRR/2020/v26i730283
Soykan, Y., Summation Formulas For Generalized Tetranacci Numbers, Asian Journal of Advanced Research and Reports, 7(2), 1-12, 2019. doi.org/10.9734/ajarr/2019/v7i230170.
Soykan, Y., Sum Formulas For Generalized Fifth-Order Linear Recurrence Sequences, Journal of Advances in Mathematics and Computer Science, 34(5), 1-14, 2019; Article no.JAMCS.53303, ISSN: 2456-9968, DOI: 10.9734/JAMCS/2019/v34i530224.
Soykan, Y., Linear Summing Formulas of Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers, Journal of Advanced in Mathematics and Computer Science, 33(3): 1-14, 2019.
Soykan, Y., On Summing Formulas of Generalized Hexanacci and Gaussian Generalized Hexanacci Numbers, Asian Research Journal of Mathematics, 14(4), 1-14, 2019; Article no.ARJOM.50727.
Soykan, Y., A Study On Sum Formulas of Generalized Sixth-Order Linear Recurrence Sequences, Asian Journal of Advanced Research and Reports, 14(2), 36-48, 2020. DOI: 10.9734/AJARR/2020/v14i230329
Soykan, Y., Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, Communications in Mathematics and Applications, 11(2), 281-295, 2020. DOI: 10.26713/cma.v11i2.1102
Soykan, Y., On Generalized Pentanacci and Gaussian Generalized Pentanacci Numbers, Asian Research Journal of Mathematics, 16(9), 102-121, 2020. DOI: 10.9734/ARJOM/2020/v16i930224
Soykan, Y., Properties of Generalized Fifth-Order Pell Numbers, Asian Research Journal of Mathematics, 15(3), 1-18, 2019.
Soykan, Y., Polatlı, E.E., A Note on Fifth Order Jacobsthal Numbers, IOSR Journal of Mathematics (IOSR-JM), 17(2), 01-23, 2021. DOI: 10.9790/5728-1702010123
Soykan, Y., A Study On Generalized 5-primes Numbers, Journal of Scientific Perspectives, 4(3), 185-202, 2020,, DOI: https://doi.org/10.26900/jsp.4.017.
Öteleş, A., Akbulak, M., A Note on Generalized k-Pell Numbers and Their Determinantal Representation, Journal of Analysis and Number Theory, 4(2), 153-158, 2016.
Waddill, M. E., The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, 9-20, 1992.
Published
2021-05-31
How to Cite
Soykan, Y. (2021). A Study on Sum Formulas of Generalized Pentanacci Sequence: Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k}. Journal of Progressive Research in Mathematics, 18(2), 20-38. Retrieved from http://www.scitecresearch.com/journals/index.php/jprm/article/view/2050
Issue
Section
Articles
Copyright (c) 2021 Journal of Progressive Research in Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
