Geodesic
On Fibonacci-like sequences
Journal of integer sequences, Tome 10 (2007) no. 10
In this note, we study Fibonacci-like sequences that are defined by the recurrence $ S_k = a, S_{k + 1} = b, S_{n + 2} \equiv S_{n + 1} + S_n (mod n + 2)$ for all $ n \geq k$, where $ k, a, b \in \mathbb{N}, 0 \leq a k, 0 \leq b k+ 1$, and $ (a,b)\ne (0,0)$. We will show that the number $ \alpha = 0.S_k S_{k+1} S_{k+2} \cdots$ is irrational. We also propose a conjecture on the pattern of the sequence $ \{S_n\}_{n \geq k}$.
Keywords: Fibonacci sequence, irrationality, mahler-type problem 
@article{JIS_2007__10_10_a2,
     author = {Vinh,  Le Anh},
     title = {On {Fibonacci-like} sequences},
     journal = {Journal of integer sequences},
     year = {2007},
     volume = {10},
     number = {10},
     zbl = {1141.05309},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_10_a2/}
}
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Vinh,  Le Anh. On Fibonacci-like sequences. Journal of integer sequences, Tome 10 (2007) no. 10. http://geodesic.mathdoc.fr/item/JIS_2007__10_10_a2/