Affiliation:
1. KONYA TECHNICAL UNIVERSITY
Abstract
In this paper, we solve the equation
\begin{equation*}
\sum_{k=0}^{m} {{2m+1}\brack{k}}_{F}\pm F_{t}=F_{n},
\end{equation*}%
under weak assumptions. Here, $F_n$ is $n^{th}$ Fibonacci number and ${{.}\brack {.}}_{F}$ denotes
Fibonomial coefficient.
Publisher
Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics
Reference13 articles.
1. Berndt, B. C., Galway, W., The Brocard–Ramanujan diophantine equation $n! + 1 = m^{2}$, Ramanujan J., 4 (2000), 41–42. https://doi.org/10.1023/A:1009873805276
2. Bollman, M., Hernandez, H. S., Luca, F., Fibonacci numbers which are sums of three factorials, Publ. Math. Debrecen, 77 (2010), 211–224.
3. Carmichael, R. D., On the numerical factors of the arithmetics forms $\alpha^{n}\pm\beta^{n}$, Annals Math., 2(15) (1913), 30-70.
4. Grossman, G., Luca, F., Sums of factorials in binary recurrence sequences, J. Number Theory, 93 (2002), 87–107. https://doi.org/10.1006/jnth.2001.2718
5. Irmak, N., Sum of the Fibonomial coefficients at most one away from Fibonacci numbers, Math. Reports, 18(68)(4) (2016), 567-571.