Krassimir T. Atanassov

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 28, 2022, Number 2, Pages 331–338

DOI: 10.7546/nntdm.2022.28.2.331-338

**Download PDF (227 Kb)**

## Details

### Authors and affiliations

**Krassimir T. Atanassov**

*Department of Bioinformatics and Mathematical Modelling,
Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
Acad. G. Bonchev Str. Bl. 105, Sofia-1113, Bulgaria
*

### Abstract

A new scheme of 2-Fibonacci sequences is introduced and the explicit formulas for its *n*-th members are given. For difference of all previous sequences from Fibonacci type, the present 2-Fibonacci sequences are obtained by a new way. It is proved that the new sequences have bases with 48 elements about function 𝜑 and modulo 9.

### Keywords

- Fibonacci sequence
- 2-Fibonacci sequence

### 2020 Mathematics Subject Classification

- 11B39

### References

- Ando, S., & Hayashi, M. (1997). Counting the number of equivalence classes of (
*m*,*F*) sequences and their generalizations.*The Fibonacci Quarterly*, 35(1), 3–8. - Atanassov, K. (1985). An arithmetical function and some of its applications.
*Bulletin of Number Theory and Related Topics*, IX(1), 18–27. - Atanassov, K. (1986). On a second new generalization of the Fibonacci sequence,
*The Fibonacci Quarterly*, 24(4), 362–365. - Atanassov, K. (2010). Combined 2-Fibonacci sequences.
*Notes on Number Theory and Discrete Mathematics*, 16(2), 24–28. - Atanassov, K. (2010). Combined 2-Fibonacci sequences. Part 2.
*Notes on Number Theory and Discrete Mathematics*, 16(4), 18–24. - Atanassov, K. (2015). A digital arithmetical function and some of its applications.

*Proceedings of the Jangjeon Mathematical Society*, 18(4), 511–528. - Atanassov, K. (2018). On two new combined 3-Fibonacci sequences.
*Notes on Number Theory and Discrete Mathematics*, 24(2), 90–93. - Atanassov, K. (2018). On two new combined 3-Fibonacci sequences. Part 2.
*Notes on Number Theory and Discrete Mathematics*, 24(3), 111–114. - Atanassov, K. (2022). On two new combined 3-Fibonacci sequences, Part 3.
*Notes on Number Theory and Discrete Mathematics*, 28(1), 143–146. - Atanassov, K., Atanassova, L., & Sasselov, D. (1985). A new perspective to the

generalization of the Fibonacci sequence.*The Fibonacci Quarterly*, 23(1), 21–28. - Atanassov, K., Atanassova, V., Shannon, A., & Turner, J. (2002).
*New Visual Perspectives on Fibonacci Numbers*. World Scientific, New Jersey. - Atanassov, K., & Shannon, A. (2016). Combined 3-Fibonacci sequences from a new type.
*Notes on Number Theory and Discrete Mathematics*, 22(3), 5–8. - Badshah, V., & Khan, I. (2009). New generalization of the Fibonacci sequence in Case of 4-order recurrence equations.
*International Journal of Theoretical & Applied Sciences*, 1(2), 93–96. - Bilgici, G. (2014). New Generalizations of Fibonacci and Lucas Sequences.
*Applied Mathematical Sciences*, 29, 1429–1437. - Dantchev, S. (1998). A closed form of the (2,
*T*) generalizations of the Fibonacci sequence,*The Fibonacci Quarterly*, 36 (5), 448–451. - Dhakne, M. B., & Godase, A. D. (2017). Properties of
*k*-Fibonacci sequence using matrix method.*MAYFEB Journal of Mathematics*, 1, 11–20. - Edson, M., Lewis, S., & Yayenie, O. (2011). The
*k*-periodic Fibonacci sequence and an Extended Binet’s formula,*Integers*, 11, 1–12. - Godase, A. D. (2017). Recurrent formulas of the generalized Fibonacci sequences of third & fourth order.
*Indian Journal in Number Theory*, 2017, 114–121. - Godase, A. D., & Dhakne, M. (2017). Identities for multiplicative coupled Fibonacci sequences of
*r*-th order.*Journal of New Theory*, 15, 48–60. - Halici, S. (2019). On Bicomplex Fibonacci Numbers and Their Generalization.
*Models and Theories in Social Systems*. Springer, Cham, 509–524. - Hirschhorn, M. (2006). Coupled third-order recurrences. Fibonacci sequence,
*The Fibonacci Quarterly*, 44(1), 20–25. - Ipek, A. (2017). On (
*p*,*q*)-Fibonacci quaternions and their Binet formulas, generating functions and certain binomial sums.*Advances in Applied Clifford Algebras*, 27(2), 1343–1351. - Kim, H. S., Neggers, J., & Park, C. (2018). On generalized Fibonacci
*k*-sequences and Fibonacci*k*-numbers.*Journal of Computational Analysis and Applications*, 24(5), 805–814. - Lee, J.-Z., & Lee, J.-S. (1987). Some properties of the generalization of the Fibonacci sequence.
*The Fibonacci Quarterly*, 25(2), 111–117. - Ömür, N. S, Koparal, S., & Sener, C. D. (2014). A New perspective to the generalization of sequences of
*t*-order.*International Journal of Computer Applications*, 86(6), 29–33. - Sikhwal, O., Vyas, Y., & Bhatnagar, S. (2017). Generalized multiplicative coupled Fibonacci sequence and its properties.
*International Journal of Computer Applications*, 158(10), 14–17. - Singh, M., Sikhwal, O., & Jain, S. (2010). Coupled Fibonacci sequences of fifth order and some properties.
*International Journal of Mathematical Analysis*, 4(25), 1247–1254. - Sisodiya, K. S., Gupta, V., & Sisodiya, K. (2014). Properties of multiplicative coupled Fibonacci sequences of fourth order under the specific schemes.
*International Journal of Mathematical Archive*, 5(4), 70–73. - Sisodiya, K. S., Gupta, V., & Sisodiya, K. (2014). Deriving a formula in solving

Fibonacci–like square sequences.*International Journal of Technology Enhancements and Emerging Engineering Research*, 2(4), 55–58. - Sisodiya, K. S., Gupta, V., & Sisodiya, K. (2014). Some fundamental properties of

multiplicative triple Fibonacci sequences.*International Journal of Latest Trends in Engineering and Technology*, 3(4), 128–131. - Spickerman, W., & Creech, R. (1997). The (2,
*T*)-generalized Fibonacci sequences.*The Fibonacci Quarterly*, 35(4), 358–360. - Spickerman, W., Creech, R., & Joyner, R. (1993). On the structure of the set of difference systems defining the (3,
*F*)-generalized Fibonacci sequence.*The Fibonacci Quarterly*, 31(4), 333–337. - Spickerman, W., Creech, R., & Joyner, R. (1995). On the (3,
*F*)-generalizations of the Fibonacci sequence.*The Fibonacci Quarterly*, 33(1), 9–12. - Spickerman, W., Creech, R., & Joyner, R. (1992). On the (2,
*F*)-generalizations of the Fibonacci sequence. The Fibonacci Quarterly, 30(4), 310–314. - Suvarnamani, A. (2017). On the multiplicative pulsating
*n*-Fibonacci sequence.*SNRU Journal of Science and Technology*, 9(2), 502–508. - Yordzhev, K. (2014). Factor-set of binary matrices and Fibonacci numbers.
*Applied Mathematics and Computation*, 236, 235–238.

### Manuscript history

- Received: 12 March 2022
- Revised: 11 May 2022
- Accepted: 11 May 2022
- Online First: 12 May 2022

## Related papers

## Cite this paper

Atanassov, K. T. (2022). Two 2-Fibonacci sequences generated by a mixed scheme. Part 1. *Notes on Number Theory and Discrete Mathematics*, 28(2), 331-338, DOI: 10.7546/nntdm.2022.28.2.331-338.