True or false: If $\alpha\in\mathbb{R}_{>1}\setminus\mathbb{N},$ then the set $\{ \lfloor \alpha^n \rfloor: n\in\mathbb{N} \}$ contains infinitely many even integers and infinitely many odd integers? I am not aware of any Pisot–Vijayaraghavan numbers for which this is false. I imagine the propo...

I have discovered an intriguing sequence defined as: If n is odd then (2^k×n+2^k) If n is even then n/2 Where k is any natural number. Starting with various positive integers, I have observed that this sequence consistently converges to 1. I would like to share a few examples: Starting with 3(k=1...