### On Generalization of Fibonacci, Lucas and Mulatu Numbers

Agung Prabowo

#### Abstract

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

#### Keywords

GFLM numbers, Fibonacci numbers, Lucas numbers, Mulatu numbers, Binet formula, Golden ratio, Silver ratio, Bronze ratio, Metallic ratio, continued fraction, nested radicals.

PDF

#### References

Bolat, C., & Köse, H. (2010). On the properties of k-Fibonacci numbers. Int. J. Contemp. Math. Sciences, 5(22), 1097-1105.

Falcón, S., and Plaza, Á. (2007). On the Fibonacci k-numbers. Chaos, Solitons & Fractals, 32(5), 1615-1624.

Fernando, G. B., and Prabowo, A. (2019) Hasil Bagi dari Jumlahan Sepuluh Bilangan Fibonacci yang Berturutan oleh 11 Adalah Bilangan Fibonacci Ketujuh, Jumlahku, 5(2), 103-110.

Kalman, D., and Mena, R. (2003). The Fibonacci numbers—exposed. Mathematics magazine, 76(3), 167-181.

Koshy, T. (2001) Fibonacci and Lucas Numbers with Applications, New York: John Wiley & Sons.

Lemma, M. (2011) The Fascinating Mathematical Beauty ofbThe Fibonacci Numbers, Proceeding HUIC-Hawaii University International Conferences on Technology and Mathematics, 1(1), 1 – 9.

Lemma, M., Lambright, J., and Atena, A. (2016). Some Fascinating Theorems Of The Mulatu Numbers. Advances and Applications in Mathematical Sciences, 15(4), 133-138.

Lemma, M. (2019). A Note on Some Interesting Theorems of the Mulatu Numbers. EPH-International Journal of Mathematics and Statistics, 5(5), 01-08.

Mulatu, L., Agegnehu, A., and Tilahun, M. (2016) The Fascinating Double Angle Formulas of the Mulatu Numbers, International Journal for Innovation Education and Research, 4(1), 25-29.

Patel, D., and Lemma, M. (2017). Using the gun of Mathematical induction to conquer some theorems of The Mulatu Numbers. Journal of Advance Research in Mathematics And Statistics, 4(3), 09-15.

Schneider, R. (2016) Fibonacci Numbers and the Golden Ratio, Parabola, 52(3), 1-6.

Sivaraman, R. (2020) Exploring Metallic Ratios, Mathematics and Statistics, 8(4), 388 – 391.

DOI: https://doi.org/10.46336/ijqrm.v1i3.65

### Refbacks

• There are currently no refbacks.

IJQRM: Jalan Riung Ampuh No. 3, Riung Bandung, Kota Bandung 40295, Jawa Barat, Indonesia

IJQRM Indexed By:

Creation is distributed below Lisensi Creative Commons Atribusi 4.0 Internasional.

View My Stats