References
- F.R. Alves and P.M. Catarino, Sequência matricial generalizada de Fibonacci e sequência matricial k-Pell: propriedades matriciais, C.Q.D. - Revista Eletrônica Paulista de Matemática 15 (2019), Edição Julho, 39–54.
- M.K. Azarian, The generating function for the Fibonacci sequence, Missouri J. Math. Sci. 2 (1990), no. 2, 78–79.
- J.B. Bacani and J.F. Rabago, On generalized Fibonacci numbers, Appl. Math. Sci. (Ruse) 9 (2015), no. 73, 3611–3622.
- G. Berzsenyi, Gaussian Fibonacci numbers, Fibonacci Quart. 15 (1977), no. 3, 233–236.
- F. Bezerra, F. Alves, and R. Vieira, Relações recorrentes bidimensionais e tridimensionais de Narayana, C.Q.D. - Revista Eletrônica Paulista de Matemática 18 (2020), Edição Julho, 12–28.
- A. Borges, P. Catarino, A. Aires, P. Vasco, and H. Campos, Two-by-two matrices involving k-Fibonacci and k-Lucas sequences, Appl. Math. Sci. (Ruse) 8 (2014), no. 34, 1659–1666.
- P. Catarino, On some identities for k-Fibonacci sequence, Int. J. Contemp. Math. Sci. 9 (2014), no. 1, 37–42.
- P. Catarino and H. Campos, From Fibonacci sequence to more recent generalisations, in: F. Yilmaz et al. (eds.), Mathematical Methods for Engineering Applications. IC-MASE 2021, Salamanca, Spain, July 1–2, Springer Proc. Math. Stat., 384, Springer, Cham, 2022, pp. 259–269.
- P. Catarino, D. Santos, and E. Costa, On t-dimensional Gersenne sequences and their symmetry properties, Symmetry 17 (2025), no. 7, Paper No. 1079, 16 pp.
- P. Catarino and P. Vasco, Some basic properties and a two-by-two matrix involving the k-Pell numbers, Int. J. Math. Anal. (Ruse) 7 (2013), no. 45, 2209–2215.
- J. Chimpanzo, P. Catarino, and M. Otero-Espinar, Some identities and generating functions for bidimensional balancing and cobalancing sequences, Univ. J. Math. Appl. 7 (2024), no. 2, 68–75.
- J. Chimpanzo, P. Catarino, and M. Otero-Espinar, Bidimensional balancing, Lucas-balancing, cobalancing and Lucas-cobalancing numbers via the determinant of a tridiagonal matrix, Indian J. Pure Appl. Math. (2025).
https://doi.org/10.1007/s13226-025-00771-z . - J. Chimpanzo, P. Catarino, P. Vasco, and A. Borges, Bidimensional extensions of balancing and Lucas-balancing numbers, J. Discrete Math. Sci. Cryptogr. 27 (2024), no. 1, 95–115.
- J. Chimpanzo, M. Otero-Espinar, A. Borges, P. Vasco, and P. Catarino, Bidimensional extensions of cobalancing and Lucas-cobalancing numbers, Ann. Math. Sil. 38 (2024), no. 2, 241–262.
- E. Costa, P. Catarino, F. Monteiro, V. Sousa, and D. Santos, Tricomplex Fibonacci numbers: a new family of Fibonacci-type sequences, Mathematics 12 (2024), no. 23, Paper No. 3723, 15 pp.
- E. Costa, P. Catarino, P. Vasco, and F. Alves, A brief study on the k-dimensional Repunit sequence, Axioms 14 (2025), no. 2, Paper No. 109, 16 pp.
- M. Edson and O. Yayenie, A new generalization of Fibonacci sequence and extended Binet's formula, Integers 9 (2009), no. 6, 639–654.
- S. Falcón, On the generating matrices of the k-Fibonacci numbers, Proyecciones 32 (2013), no. 4, 347–357.
- S. Falcón, On the extended (k, t)-Fibonacci numbers, J. Adv. Math. Comput. Sci. 39 (2024), no. 7, 81–89.
- S. Falcón and Á. Plaza, On the Fibonacci k-numbers, Chaos Solitons Fractals 32 (2007), no. 5, 1615–1624.
- H.W. Gould, A history of the Fibonacci Q-matrix and a higher-dimensional problem, Fibonacci Quart. 19 (1981), no. 3, 250–257.
- C.J. Harman, Complex Fibonacci numbers, Fibonacci Quart. 19 (1981), no. 1, 82–86.
- A.F. Horadam, A generalized Fibonacci sequence, Amer. Math. Monthly 68 (1961), no. 5, 455–459.
- A.F. Horadam, Complex Fibonacci numbers and Fibonacci quaternions, Amer. Math. Monthly 70 (1963), no. 3, 289–291.
- A.F. Horadam, Basic properties of a certain generalized sequence of numbers, Fibonacci Quart. 3 (1965), no. 3, 161–176.
- H. Hosoya, Fibonacci triangle, Fibonacci Quart. 14 (1976), no. 2, 173–179.
- J.H. Jordan, Gaussian Fibonacci and Lucas numbers, Fibonacci Quart. 3 (1965), no. 4, 315–318.
- D. Kalman and R. Mena, The Fibonacci numbers–exposed, Math. Mag. 76 (2003), no. 3, 167–181.
- C.H. King, Some Properties of the Fibonacci Numbers, Master's thesis, San Jose State College, 1960.
- C. Kızılateş, P. Catarino, and N. Tuğlu, On the bicomplex generalized Tribonacci quaternions, Mathematics 7 (2019), no. 1, Paper No. 80, 8 pp.
- T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 1, John Wiley & Sons, Hoboken, NJ, 2017.
- B. Kuloğlu and E. Özkan, On the (p, q)-Narayana n-dimensional recurrences, J. Sci. Arts 23 (2023), no. 3, 707–714.
- R. Oliveira and F. Alves, Os números Gaussianos de Fibonacci e relações recorrentes bidimensionais, Rev. Thema 16 (2019), no. 4, 745–754.
- R. Oliveira, F. Alves, and R. Paiva, Identidades bi e tridimensionais para os números de Fibonacci na forma complexa, C.Q.D. - Revista Eletrônica Paulista de Matemática 11 (2017), Edição Dezembro, 91–106.
- S. Pethe and A.F. Horadam, Generalised Gaussian Fibonacci numbers, Bull. Aust. Math. Soc. 33 (1986), no. 1, 37–48.
- B. Singh, O. Sikhwal, and Y. Gupta, Generalized Fibonacci–Lucas sequence, Turk. J. Anal. Number Theory 2 (2014), no. 6, 193–197.
- N.J. Sloane et al., The On-Line Encyclopedia of Integer Sequences, The OEIS Foundation Inc.,
https://oeis.org . - S. Uygun, Complex Jacobsthal numbers in two dimension, Sarajevo J. Math. 20(33) (2024), no. 2, 219–229.
- S. Vajda, Fibonacci and Lucas Numbers, and the Golden Section: Theory and Applications, Courier Corporation, Chelmsford, MA, 2008.
- R. Vieira, F. Alves, and P. Catarino, Relações bidimensionais e identidades da sequência de Leonardo, Rev. Sergipana Mat. Educ. Mat. 4 (2019), no. 2, 156–173.
- A. Wani, V. Badshah, G. Rathore, and P. Catarino, Generalized Fibonacci and k-Pell matrix sequences, Punjab Univ. J. Math. (Lahore) 51 (2020), no. 1, 17–28.
- O. Yayenie, A note on generalized Fibonacci sequences, Appl. Math. Comput. 217 (2011), no. 12, 5603–5611.