AMM Problem

## AMM Problem 11651

Marcel Celaya, Department of Computer Science, University of Victoria.
Frank Ruskey, Department of Computer Science, University of Victoria.

### Abstract:

Show that the equality $\left\lfloor \frac{n+1}{\phi} \right\rfloor = n - {\small{\left\lfloor \frac{n}{\phi} \right\rfloor}} + \biggl\lfloor \frac{ \bigl\lfloor \frac{n}{\phi} \bigr\rfloor}{\phi} \biggr\rfloor - \biggl\lfloor \frac{ \bigl\lfloor \frac{\left\lfloor \frac{n}{\phi} \right\rfloor}{\phi} \bigr\rfloor}{\phi} \biggr\rfloor + \Biggl\lfloor \frac{ \biggl\lfloor \frac{ \bigl\lfloor \frac{\left\lfloor \frac{n}{\phi} \right\rfloor}{\phi} \bigr\rfloor}{\phi} \biggr\rfloor}{\phi} \Biggr\rfloor - \cdots$ holds for all non-negative integers $n$ if and only if $\phi = (1+\sqrt{5})/2$, the golden ratio.