On 5/24/08, Fred lunnon <fred.lunnon@gmail.com> wrote:
[There must be something about this expression --- the version below is/was wrong in OEIS too!]
Apologies to OEIS --- the Fibonacci formula _is_ correct there --- I just need a new pair of spectacles --- or maybe a new pair of eyes --- new brain --- witter, witter ... Speaking of which --- coincidentally with RCS revealing unsuspected depths in the intger part function --- I've turned up a number of apparently elementary identities for which I can discover no obvious proof technique. The following example [occurring in the investigation of the "shifted down 2" variation of the Knuth circle product] is typical: for integer x, x - [[x/tau^2 + 1/tau^2]*tau^2 + 1/tau] = ( - ([(x+3)*tau^2] - 2*[(x+2)*tau^2] + [(x+1)*tau^2]) )mod 3 - 1 The sequence concerned, starting from x = 0, is 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 0, 1, ... The second expression is much easier to deal with than the first, since the iterated integer part has been eliminated. But why are the two equal? Fred Lunnon