Re: [math-fun] Knuth circle product [was ... Zeckendorf representation]

On 5/27/08, Fred lunnon <fred.lunnon@gmail.com> wrote:

I now have proofs of both

x o y = 3 x y - x [(y+1)/phi^2] - y [(x+1)/phi^2]

and

x @ y = x y - [(y+1)/phi^2] [(x+1)/phi^2]

and (x o y) o z = 8 x y z - 3 (x' y z + x y' z + x y z') + (x y' z' + x' y z' + x' y' z) where x', y', z' denote [(x+1)/phi^2], [(y+1)/phi^2], [(z+1)/phi^2] . Since this is symmetric in x,y,z, it provides an alternative proof of Knuth's theorem that x o y is associative. Given that x @ y is not associative --- so can have no such triple product formula --- it's a little surprising that the above drops out so nicely! Fred Lunnon