On 5/17/08, N. J. A. Sloane <njas@research.att.com> wrote:

WFL said on May 16: PS --- Fibonacci product table, as defined using base-F' above: ... Me: I am reading these messages in reverse order. This looks like an interesting array that should be in the OEIS (converted to a sequence by being read by antidaigonals). Fred, is it correct? If so, could you please linearize it and send it in the OEIS along with a definition? (In return for my telling you about A101330!)

The variant table is identical to Knuth's original, except for redefining x (*) 0 = 0 (*) x = x instead of x (*) 0 = 0 (*) x = 0. I suggest a footnote in A101330 mentioning that this also results in an associative operation --- but it's almost certainly the "wrong" variation, because it conflicts with known explicit formulae for x (*) y when x is constant. This variant results from specifying that the coefficient of F_0 in the representation of 0 (alone) shall be 1 instead of 0, i.e. 0 = ...0001 instead of ...0000. I've looked at similar plausibilities, and the only other associative variant I've found involves representing (alone) 1 = ...0011 instead of ...0100, which gives the same "circle" product as the original. Fred Lunnon