Generalized John Numbers
Keywords:
John numbers, John-Lucas numbers, Tribonacci numbers, Pell numbers, Pell-Lucas numbers.
Abstract
In this paper, we define and investigate the generalized John sequences and we deal with, in detail, two special cases, namely, John and John-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 John and John-Lucas numbers and Pell, Pell-Lucas numbers.
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References
Bicknell, N., A primer on the Pell sequence and related sequence, Fibonacci Quarterly, 13(4), 345-349, 1975.
Dasdemir, A., On the Pell, Pell-Lucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences, 5(64), 3173-3181, 2011.
Ercolano, J., Matrix generator of Pell sequence, Fibonacci Quarterly, 17(1), 71-77, 1979.
Gökbas, 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.
Horadam, A. F., Pell identities, Fibonacci Quarterly, 9(3), 245-263, 1971.
Kiliç, E., Taşçi, D., The Linear Algebra of The Pell Matrix, Boletín de la Sociedad Matemática Mexicana, 3(11), 2005.
Kiliç, E., Taşçi, D., The Generalized Binet Formula, Representation and Sums of the Generalized Order-k Pell Numbers, Taiwanese Journal of Mathematics, 10(6), 1661-1670, 2006.
Kiliç, E., Stanica, P., A Matrix Approach for General Higher Order Linear Recurrences, Bulletin of the Malaysian Mathematical Sciences Society, (2) 34(1), 51--67, 2011.
Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
Melham, R., Sums Involving Fibonacci and Pell Numbers, Portugaliae Mathematica, 56(3), 309-317, 1999.
N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
Soykan, Y., On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports, 6(1): 1-18, 2019.
Soykan, Y., A Study of Generalized Fourth-Order Pell Sequences, Journal of Scientific Research and Reports, 25(1-2), 1-18, 2019.
Soykan, Y., Properties of Generalized Fifth-Order Pell Numbers, Asian Research Journal of Mathematics, 15(3), 1-18, 2019.
Soykan, Y., On Generalized Sixth-Order Pell Sequence, Journal of Scientific Perspectives, 4(1), 49-70, 2020, DOİ: https://doi.org/10.26900/jsp.4.005.
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), 45-56, 2019.
Soykan Y., A Study On Generalized (r,s,t)-Numbers, MathLAB Journal, 7, 101-129, 2020.
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
Soykan, Y., Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{mk+j}, Archives of Current Research International, 21(3), 11-38, 2021. DOI: 10.9734/ACRI/2021/v21i330235
Dasdemir, A., On the Pell, Pell-Lucas and Modified Pell Numbers By Matrix Method, Applied Mathematical Sciences, 5(64), 3173-3181, 2011.
Ercolano, J., Matrix generator of Pell sequence, Fibonacci Quarterly, 17(1), 71-77, 1979.
Gökbas, 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.
Horadam, A. F., Pell identities, Fibonacci Quarterly, 9(3), 245-263, 1971.
Kiliç, E., Taşçi, D., The Linear Algebra of The Pell Matrix, Boletín de la Sociedad Matemática Mexicana, 3(11), 2005.
Kiliç, E., Taşçi, D., The Generalized Binet Formula, Representation and Sums of the Generalized Order-k Pell Numbers, Taiwanese Journal of Mathematics, 10(6), 1661-1670, 2006.
Kiliç, E., Stanica, P., A Matrix Approach for General Higher Order Linear Recurrences, Bulletin of the Malaysian Mathematical Sciences Society, (2) 34(1), 51--67, 2011.
Koshy, T., Pell and Pell-Lucas Numbers with Applications, Springer, New York, 2014.
Melham, R., Sums Involving Fibonacci and Pell Numbers, Portugaliae Mathematica, 56(3), 309-317, 1999.
N.J.A. Sloane, The on-line encyclopedia of integer sequences, http://oeis.org/
Soykan, Y., On Generalized Third-Order Pell Numbers, Asian Journal of Advanced Research and Reports, 6(1): 1-18, 2019.
Soykan, Y., A Study of Generalized Fourth-Order Pell Sequences, Journal of Scientific Research and Reports, 25(1-2), 1-18, 2019.
Soykan, Y., Properties of Generalized Fifth-Order Pell Numbers, Asian Research Journal of Mathematics, 15(3), 1-18, 2019.
Soykan, Y., On Generalized Sixth-Order Pell Sequence, Journal of Scientific Perspectives, 4(1), 49-70, 2020, DOİ: https://doi.org/10.26900/jsp.4.005.
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), 45-56, 2019.
Soykan Y., A Study On Generalized (r,s,t)-Numbers, MathLAB Journal, 7, 101-129, 2020.
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
Soykan, Y., Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{mk+j}, Archives of Current Research International, 21(3), 11-38, 2021. DOI: 10.9734/ACRI/2021/v21i330235
Published
2022-03-15
How to Cite
Soykan, Y. (2022). Generalized John Numbers. Journal of Progressive Research in Mathematics, 19(1), 17-34. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2124
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