Sum Formulas For Generalized Tetranacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k}
Keywords:
Tetranacci numbers, Tetranacci-Lucas numbers, fourth order Pell numbers, sum formulas, summing formulas.
Abstract
In this paper, closed forms of the sum formulas ∑_{k=0}ⁿx^{k}W_{k} and ∑_{k=1}ⁿx^{k}W_{-k} for generalized Tetranacci numbers are presented. As special cases, we give summation formulas of Tetranacci, Tetranacci-Lucas, and other fourth-order recurrence sequences.
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References
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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.
Hathiwala, G. S., Shah, D. V., Binet--Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods, Mathematical Journal of Interdisciplinary Sciences 6 (1), 37--48, 2017.
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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.
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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
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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., Gaussian Generalized Tetranacci Numbers, Journal of Advances in Mathematics and Computer Science, 31(3): 1-21, Article no.JAMCS.48063, 2019.
Soykan, Y., A Study of Generalized Fourth-Order Pell Sequences, Journal of Scientific Research and Reports, 25(1-2), 1-18, 2019.
Polatlı, E.E., Soykan, Y., A Study on Generalized Fourth-Order Jacobsthal Sequences, Submitted.
Soykan, Y., On Generalized 4-primes Numbers, Int. J. Adv. Appl. Math. and Mech. 7(4), 20-33, 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
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.
Waddill, Another Generalized Fibonacci Sequence, M. E., Fibonacci Quarterly, 5 (3), 209-227, 1967.
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.
Hathiwala, G. S., Shah, D. V., Binet--Type Formula For The Sequence of Tetranacci Numbers by Alternate Methods, Mathematical Journal of Interdisciplinary Sciences 6 (1), 37--48, 2017.
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.
Singh, B., Bhadouria, P., Sikhwal, O., Sisodiya, K., A Formula for Tetranacci-Like Sequence, Gen. Math. Notes, 20 (2), 136-141, 2014.
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., Gaussian Generalized Tetranacci Numbers, Journal of Advances in Mathematics and Computer Science, 31(3): 1-21, Article no.JAMCS.48063, 2019.
Soykan, Y., A Study of Generalized Fourth-Order Pell Sequences, Journal of Scientific Research and Reports, 25(1-2), 1-18, 2019.
Polatlı, E.E., Soykan, Y., A Study on Generalized Fourth-Order Jacobsthal Sequences, Submitted.
Soykan, Y., On Generalized 4-primes Numbers, Int. J. Adv. Appl. Math. and Mech. 7(4), 20-33, 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
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.
Waddill, Another Generalized Fibonacci Sequence, M. E., Fibonacci Quarterly, 5 (3), 209-227, 1967.
Published
2021-02-19
How to Cite
Soykan, Y. (2021). Sum Formulas For Generalized Tetranacci Numbers: 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(1), 24-47. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2018
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