If corresponding factorials and triangular numbers are added, the results form the sequence of numbers, {2, 5, 12, 34, 135, 741, 5068, 40356, ...}, which I call factoriangular numbers. In the list of the first few factoriangular numbers, I found three Fibonacci numbers: 2, 5 and 34. Aside from these three, are there other Fibonacci numbers in the sequence of factoriangular numbers?
Dear Romer,
On the direction of the problem that you are proposing, the only result I know is due to Surajit Rajagopal and Martin Griffiths, that showed when Fibonacci numbers are also factorial numbers. (http://dx.doi.org/10.2307/3621818)
Cheers,
Ana Paula
Dear Romer,
You can use The On-Line Encyclopedia of Integer Sequences® (OEIS®)
https://oeis.org
Sincerelly Yuri
A very recent research paper of Ruiz and Luca proved that 2, 5 and 34 are the only Fibonacci factoriangular numbers. See: https://doi.org/10.1016/j.indag.2017.05.002.
The article cited in the said paper of Ruiz and Luca is given below:
Article On the Sum of Corresponding Factorials and Triangular Number...
Other related papers that may interest the readers of this thread are as follows:
Article On the Sum of Corresponding Factorials and Triangular Number...
Article Sums of Two Triangulars and of Two Squares Associated with S...
Article A Survey on Triangular Number, Factorial and Some Associated Numbers
Article Recurrence Relations and Generating Functions of the Sequenc...
In 2017, Ruiz, Luca and Togbe proved this result in the article "Fibonacci factoriangular numbers".
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