Is there a way to describe the sequence $2,2,1,0,0,1,2,2,1,0,0,1,2,2...$ by using the floor function?
I can describe the series using a sinusoidal function but wanted to get it in terms of a floor function for a tricky proof. The sinusoidal function is:
$u(n)=-\frac{2\sqrt{3}}{3}sin(\frac{\pi}{3}n-\pi)+1$
$\lfloor(n+5)/6\rfloor-\lfloor(n+4)/6\rfloor$ gives the sequence $1,0,0,0,0,0$ repeating. So what you want is $$2\lfloor{n+5\over6}\rfloor-2\lfloor{n+4\over6}\rfloor+2\lfloor{n+4\over6}\rfloor-2\lfloor{n+3\over6}\rfloor+\lfloor{n+3\over6}\rfloor-\lfloor{n+2\over6}\rfloor+\lfloor{n\over6}\rfloor-\lfloor{n-1\over6}\rfloor$$ which simplifies a little to $$2\lfloor{n+5\over6}\rfloor-\lfloor{n+3\over6}\rfloor-\lfloor{n+2\over6}\rfloor+\lfloor{n\over6}\rfloor-\lfloor{n-1\over6}\rfloor$$