0

The famous Fibonacci sequence is defined by the recurrence relation $$F_n=F_{n-1}+F_{n-2}$$ where $F_n$ is the $n^{\text{th}}$ term of the sequence. What if instead of adding the previous two terms, we multiply them? So the recurrence relation becomes $$F_n^*=F_{n-1}^*×F_{n-2}^*$$ where $F_n^*$ is the $n^{\text{th}}$ term of the sequence. The usual initial terms $0$ and $1$ gives a boring sequence, but other initial terms like $2$ and $3$ gives something better.

Now what kind of properties would this sequence have? When I tried googling it, I had only found this math paper and this MSE post. Is there any other information about this?

  • 2
    Look a this. – TheSilverDoe Jul 06 '23 at 14:18
  • 6
    Just take the logarithm to get an additive Fibonacci-like sequence again. – See https://math.stackexchange.com/q/3314911/42969 about the Fibonacci sequence with arbitrary start values. – Martin R Jul 06 '23 at 14:19
  • @TheSilverDoe Thanks for mentioning it. I had seen that before, but forgot to mention –  Jul 06 '23 at 14:24
  • Just keep count of the exponents of $2$ and $3$: $F_n^* = 2^{F_{n-1}}3^{F_n}$. Did you even try a little to solve the problem? – jjagmath Jul 06 '23 at 15:27
  • @jjagmath Yes, I had noticed that. This is not a problem, I just asked for some interesting properties. I had found some properties myself too. –  Jul 06 '23 at 15:33
  • 1
    @HishamShefeekh Then you should share your findings instead of expecting others to redo your work. – jjagmath Jul 06 '23 at 15:35

0 Answers0