A New Family of Number Sequences: Leonardo-Alwyn Numbers
DOI:
https://doi.org/10.52737/18291163-2023.15.6-1-13Keywords:
Leonardo Number, Leonardo-Alwyn Number, John-Edouard Number, Ernst NumberAbstract
In this study, we define a new type of number sequence called Leonardo-Alwyn sequence. We obtain the Binet formula, generating function and some relations for these numbers. Moreover, we give the matrix representation of the Leonardo-Alwyn numbers.
References
S.L. Adler, Quaternionic quantum mechanics and quantum fields, New York Oxford University Press, 1994.
F. Alves and R. Vieira, The Newton fractal’s Leonardo sequence study with the Google Colab. Int. Elect. J. Math. Ed., 15 (2020), no. 2, pp. 1-9. https://doi.org/10.29333/iejme/6440
J. Baez, The octonians. Bull. Amer. Math. Soc., 145 (2001), no. 39, pp. 145-205.
J. Bravo, C.A. Gomez and J. L. Herrera, On the intersection of k-Fibonacci and Pell numbers. Bull. Korean Math. Soc., 56 (2019), no. 2, pp. 535-547.
P. Catarino and A. Borges, On Leonardo numbers. Acta Math. Univ. Comenianae, 1 (2020), pp. 75-86.
A. Daşdemir, On the Jacobsthal numbers by matrix method. SDU Journal of Science, 7 (2012), no. 1, pp. 69-76.
A. Dubbs and A. Edelman, Infinite random matrix theory, tridiagonal bordered Toeplitz matrices, and the moment problem. Linear Algebra Appl., 467 (2015), pp. 188-201. https://doi.org/10.1016/j.laa.2014.11.006
M. Edson and O. Yayenie, A new generalization of Fibonacci sequences and the extended Binet’s formula. Inyegers Electron. J. Comb. Number Theory, 9 (2009), no. 6, pp. 639-654. https://doi.org/10.1515/integ.2009.051
S. Falcón and Á. Plaza, On the Fibonacci k-numbers. Chaos, Solitons and Fractals, 32 (2007), no. 5, pp. 1615-1624. https://doi.org/10.1016/j.chaos.2006.09.022
S. Falcón, The k-Fibonacci difference sequences. Chaos, Solitons and Fractals, 87 (2016), pp. 153-157. https://doi.org/10.1016/j.chaos.2016.03.038
A.H. George, Some formula for the Fibonacci sequence with generalization. Fibonacci Q., 7 (1969), pp. 113-130.
W.R. Hamilton, Li on quaternions; or on a new system of Imaginaries in algebra. Philos. Mag. Ser. Taylor and Francis, 25 (1844), no. 163.
C.J. Harman, Complex Fibonacci numbers. Fibonacci Q., 19 (1981), no. 1, pp. 82-86.
A. Heeffer and A.M. Hinz, A difficult case: Pacioli and Cardano on the Chinese rings. Recreat. Math. Mag., 4 (2017), no. 8, pp. 5-23. https://doi.org/10.1515/rmm-2017-0017
A.M. Hinz, S. Klavzar and C. Petr, The tower of Hanoi-Myths and maths, Birkhauser, 2018.
A.F. Horadam, A generalized Fibonacci sequence. Math. Mag., 68 (1961), pp. 455-459.
A.F. Horadam, Basic properties of a certain generalized sequence of numbers. Fibonacci Q., 3 (1965), pp. 161-176.
R.E. Hudson, C.W. Reed, D. Chen and F. Lorenzelli, Blind beamforming on a randomly distributed sensor array system. J. Sel. Areas Commun., 16 (1998), no. 8, pp. 1555-1567. https://doi.org/10.1109/49.730461
B. Kafle, S.E. Rihane and A. Togbe, Pell and Pell-Lucas numbers of the form $x^{a} ± x^{b}+1$. Bol. Soc. Mat. Mex., 26 (2020), pp. 879-893. https://doi.org/10.1007/s40590-020-00305-z
T. Koshy, Fibonacci and Lucas numbers with applications, Wiley, 2001.
T. Koshy, Pell and Pell-Lucas numbers with applications. In: Pell and Pell–Lucas Numbers with Applications, Springer, New York, NY, 2014. https://doi.org/10.1007/978-1-4614-8489-9_7
D. Marques and P. Trojovsky, On characteristic polynomial of higher order generalized Jacobsthal numbers. Advances in Continuous and Discrete Models, 2019 (2019), no. 392, pp. 1-9. https://doi.org/10.1186/s13662-019-2327-6
E. Ngondiep, S. Serra-Capizzano and D. Sesana, Spectral features and asymptotic properties for g-circulant and g-Toeplitz sequence. SIAM J. Mat. Anal. App., 31 (2010), no. 4, pp. 1663-1687. https://doi.org/10.1137/090760209
L. Ramirez, Some combinatorial properties of the k-Fibonacci and the k-Lucas quaternions. An. St. Univ. Ovidius Constanta, Ser. Mat., 23 (2015), no. 2, pp. 201-212. https://doi.org/10.1515/auom-2015-0037
S.F. Santana and J.L. Barrero, Some properties of sums involving Pell numbers. Missouri J. Math. Sci., 18 (2006), no. 1, pp. 33-40. https://doi.org/10.35834/2006/1801033
Y. Soykan, On k-circulant matrices with the generalized third-order Pell numbers. NNTDM, 27 (2021), no. 4, pp. 187-206. https://doi.org/10.7546/nntdm.2021.27.4.187-206
L. Spelina and I. Wolch, On generalized Pell and Pell-Lucas numbers. Iranian J. Sci. Tech., 43 (2019), pp. 2871-2877. https://doi.org/10.1007/s40995-019-00757-7
Ş. Uygun and E. Owusu, A new generalization of Jacobsthal Lucas numbers. J. Adv. Math. Comp. S., 34 (2019), no. 5, pp. 1-13.
S. Vajda, Fibonacci and Lucas numbers and the golden section, Ellis Horwood Limited Publ., England, 1989.
R. Vieira, M. Mangueira, F. Alves and P. Catarino, A forma matricial dos números de Leonardo. Ciência E Natura, 42 (2020), pp. 1-12. https://doi.org/10.5902/2179460x41839
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