Generalized Leonardo Numbers
Keywords:
Leonardo numbers, Leonardo-Lucas numbers, Tribonacci numbers, Fibonacci numbers, Lucas numbers.
Abstract
In this paper, we investigate the generalized Leonardo sequences and we deal with, in detail, three special cases, namely, modified Leonardo, Leonardo-Lucas and Leonardo 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 modified Leonardo, Leonardo-Lucas, Leonardo numbers and Fibonacci, Lucas numbers.
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References
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Bruce, I., A modified Tribonacci sequence, Fibonacci Quarterly, 22(3), 244--246, 1984.
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P, Catarino., Borges, A., A Note on Incomplete Leonardo Numbers, Integers, Vol:20, 2020.
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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.
Kalman, D., Generalized Fibonacci Numbers By Matrix Methods, Fibonacci Quarterly, 20(1), 73-76, 1982.
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Pethe, S., Some Identities for Tribonacci sequences, Fibonacci Quarterly, 26(2), 144--151, 1988.
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Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), 45-56, 2019.
Soykan, Y. Tribonacci and Tribonacci-Lucas Sedenions. Mathematics 7(1), 74, 2019.
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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
Soykan, Y., Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: the Sum Formula ∑_{k=0}ⁿx^{k}W_{mk+j}², Journal of Advances in Mathematics and Computer Science, 36(6), 30-62, 2021; ISSN: 2456-9968. DOI: 10.9734/JAMCS/2021/v36i630371
Soykan, Y., Sums of Cubes of Generalized Fibonacci Numbers with Indices in Arithmetic Progression: the Sum Formulas ∑_{k=0}ⁿx^{k}W_{mk+j}³, Int. J. Adv. Appl. Math. and Mech. 9(1), 6-41, 2021.
Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
Vieira, R.P.M., Alves F.R.V., Catarino, P.M., Relacoes Bidimensionais E Identidades Da Sequencia De Leonardo, Revista Sergipana de Matematica e Educaçao Matematica, 156-173, 2019. https://doi.org/10.34179/revisem.v4i2.11863
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.
Bruce, I., A modified Tribonacci sequence, Fibonacci Quarterly, 22(3), 244--246, 1984.
Catalani, M., Identities for Tribonacci-related sequences, arXiv:math/0209179, 2012.
P, Catarino., Borges, A., A Note on Incomplete Leonardo Numbers, Integers, Vol:20, 2020.
P, Catarino., Borges, A., On Leonardo Numbers, Acta Mathematica Universitatis Comenianae, 89(1), 75--86, 2019.
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.
Kalman, D., Generalized Fibonacci Numbers By Matrix Methods, Fibonacci Quarterly, 20(1), 73-76, 1982.
Lin, P. Y., De Moivre-Type Identities For The Tribonacci Numbers, Fibonacci Quarterly, 26, 131-134, 1988.
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.
Shannon, A.G., A Note On Generalized Leonardo Numbers, Notes on Number Theory and Discrete Mathematics, Vol:25, No. 3 (2019), 97-101.
Sloane, N.J.A., The on-line encyclopedia of integer sequences, http://oeis.org/
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), 45-56, 2019.
Soykan, Y. Tribonacci and Tribonacci-Lucas Sedenions. Mathematics 7(1), 74, 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
Soykan, Y., Generalized Fibonacci Numbers with Indices in Arithmetic Progression and Sum of Their Squares: the Sum Formula ∑_{k=0}ⁿx^{k}W_{mk+j}², Journal of Advances in Mathematics and Computer Science, 36(6), 30-62, 2021; ISSN: 2456-9968. DOI: 10.9734/JAMCS/2021/v36i630371
Soykan, Y., Sums of Cubes of Generalized Fibonacci Numbers with Indices in Arithmetic Progression: the Sum Formulas ∑_{k=0}ⁿx^{k}W_{mk+j}³, Int. J. Adv. Appl. Math. and Mech. 9(1), 6-41, 2021.
Spickerman, W., Binet's formula for the Tribonacci sequence, Fibonacci Quarterly, 20, 118--120, 1982.
Vieira, R.P.M., Alves F.R.V., Catarino, P.M., Relacoes Bidimensionais E Identidades Da Sequencia De Leonardo, Revista Sergipana de Matematica e Educaçao Matematica, 156-173, 2019. https://doi.org/10.34179/revisem.v4i2.11863
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.
Published
2021-11-23
How to Cite
Soykan, Y. (2021). Generalized Leonardo Numbers. Journal of Progressive Research in Mathematics, 18(4), 58-84. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2101
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