A Study on Sum Formulas for Generalized Tribonacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k}
Keywords:
Tribonacci numbers, Padovan numbers, Perrin numbers, sum formulas
Abstract
In this paper, closed forms of the sum formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k} for generalized Tribonacci numbers are presented. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third order linear recurrance sequences.
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References
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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.
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.
Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
Parpar, T., k'ncı Mertebeden Rekürans Bağıntısının Özellikleri ve Bazı Uygulamaları, Selçuk Üniversitesi, Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 2011.
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.G, Horadam, A.F., Some Properties of Third-Order Recurrence Relations, The Fibonacci Quarterly, 10 (2), 135-146, 1972.
Shannon, A., Tribonacci numbers and Pascal's pyramid, Fibonacci Quarterly, 15(3), pp. 268 and 275, 1977.
Soykan, Y., On Summing Formulas For Generalized Fibonacci and Gaussian Generalized Fibonacci Numbers, Advances in Research, 20(2), 1-15, 2019.
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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., Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, arXiv:1809.07809v1 [math.NT] 20 Sep 2018.
Soykan, 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., 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., 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., 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., 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., 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., On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports, 6(1): 1-18, 2019.
Soykan Y., On Generalized Padovan Numbers, MathLAB Journal, In Print.
Soykan, Y., Generalized Pell-Padovan Numbers, Asian Journal of Advanced Research and Reports, 11(2), 8-28, 2020. DOI: 10.9734/AJARR/2020/v11i230259
Soykan Y., On Generalized Narayana Numbers, Int. J. Adv. Appl. Math. and Mech. 7(3), 43-56, 2020, (ISSN: 2347-2529).
Soykan, Y., A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences, 4(2), 227-251, 2020. https://doi.org/10.34198/ejms.4220.227251
Polatlı, E.E., Soykan, Y., On Generalized Third-Order Jacobsthal Numbers, Submitted.
Soykan, Y., On Generalized Grahaml Numbers, Journal of Advances in Mathematics and Computer Science, 35(2), 42-57, 2020. DOI: 10.9734/JAMCS/2020/v35i230248.
Soykan Y., On Generalized Reverse 3-primes Numbers, Journal of Scientific Research and Reports, 26(6), 1-20, 2020. DOI: 10.9734/JSRR/2020/v26i630267
Soykan Y., Generalized Tribonacci Numbers: Summing Formulas, Int. J. Adv. Appl. Math. and Mech. 7(3), 57-76, 2020.
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
N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
Yalavigi, C.C., A Note on `Another Generalized Fibonacci Sequence', The Mathematics Student. 39, 407--408, 1971.
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.
Marcellus E. Waddill, 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.
Waddill, M. E., The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, 9-20, 1992.
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.
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.
Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
Parpar, T., k'ncı Mertebeden Rekürans Bağıntısının Özellikleri ve Bazı Uygulamaları, Selçuk Üniversitesi, Fen Bilimleri Enstitüsü, Yüksek Lisans Tezi, 2011.
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.G, Horadam, A.F., Some Properties of Third-Order Recurrence Relations, The Fibonacci Quarterly, 10 (2), 135-146, 1972.
Shannon, A., Tribonacci numbers and Pascal's pyramid, Fibonacci Quarterly, 15(3), pp. 268 and 275, 1977.
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, 2020. https://www.researchgate.net/publication/337063487_Corrigendum_On_Summing_Formulas
_For_Generalized_Fibonacci_and_Gaussian_Generalized_Fibonacci_Numbers.
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., Matrix Sequences of Tribonacci and Tribonacci-Lucas Numbers, arXiv:1809.07809v1 [math.NT] 20 Sep 2018.
Soykan, 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., 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., 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., 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., 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., 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., On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports, 6(1): 1-18, 2019.
Soykan Y., On Generalized Padovan Numbers, MathLAB Journal, In Print.
Soykan, Y., Generalized Pell-Padovan Numbers, Asian Journal of Advanced Research and Reports, 11(2), 8-28, 2020. DOI: 10.9734/AJARR/2020/v11i230259
Soykan Y., On Generalized Narayana Numbers, Int. J. Adv. Appl. Math. and Mech. 7(3), 43-56, 2020, (ISSN: 2347-2529).
Soykan, Y., A Study on Generalized Jacobsthal-Padovan Numbers, Earthline Journal of Mathematical Sciences, 4(2), 227-251, 2020. https://doi.org/10.34198/ejms.4220.227251
Polatlı, E.E., Soykan, Y., On Generalized Third-Order Jacobsthal Numbers, Submitted.
Soykan, Y., On Generalized Grahaml Numbers, Journal of Advances in Mathematics and Computer Science, 35(2), 42-57, 2020. DOI: 10.9734/JAMCS/2020/v35i230248.
Soykan Y., On Generalized Reverse 3-primes Numbers, Journal of Scientific Research and Reports, 26(6), 1-20, 2020. DOI: 10.9734/JSRR/2020/v26i630267
Soykan Y., Generalized Tribonacci Numbers: Summing Formulas, Int. J. Adv. Appl. Math. and Mech. 7(3), 57-76, 2020.
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
N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
Yalavigi, C.C., A Note on `Another Generalized Fibonacci Sequence', The Mathematics Student. 39, 407--408, 1971.
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.
Marcellus E. Waddill, 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.
Waddill, M. E., The Tetranacci Sequence and Generalizations, Fibonacci Quarterly, 9-20, 1992.
Published
2020-10-22
How to Cite
Soykan, Y. (2020). A Study on Sum Formulas for Generalized Tribonacci Numbers: Closed Forms of the Sum Formulas ∑_{k=0}ⁿkx^{k}W_{k}, ∑_{k=1}ⁿkx^{k}W_{-k}. Journal of Progressive Research in Mathematics, 17(1), 1-40. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/1934
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