I proved an interesting pattern involving the golden ratio, for which I couldn’t find any reference online, so I just wanted to see if people could provide me with a reference, or a proof for the following: (My proof is quite indirect and so I’m hoping for something more direct)
Let $\phi$ denote the golden ratio, that is, the positive root of the quadratic $x^2-x+1=0$. Consider the two functions defined on the set of positive integers:
$$f(n)=\lfloor \phi n\rfloor$$ $$g(n) =\lfloor \phi^2 n \rfloor,$$
where $\lfloor.\rfloor$ denotes the greatest integer function. I claim that that the ranges of these two functions provide a partition of the set of positive integers. That is, every positive integer occurs in the range of exactly one of these functions.
My approach is to derive some kind of functional equations that are satisfied by these functions, but the argument gets pretty long-winded.
This is a special case of Beatty's Theorem, which states that, if $\alpha$ and $\beta$ are positive irrational numbers satisfying $\frac1\alpha+\frac1\beta=1$, the sequences $\{\lfloor\alpha n\rfloor:n\in\mathbb N\}$ and $\{\lfloor\beta n\rfloor:n\in \mathbb N\}$ partition $\mathbb N$. There are many proofs of this theorem, two of which are given in the linked Wikipedia article (and many more of which can be found online).