You might begin by changing powers of tau into a*tau+b, and moving everything that identifiably an integral term outside the floor brackets. I doubt that's the whole answer, but it could simplify your problem. Rich ---------- Quoting Fred lunnon <fred.lunnon@gmail.com>:
On 5/25/08, Fred lunnon <fred.lunnon@gmail.com> wrote:
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?
PS As previously, tau = (1 + sqrt(5))/2 and [...] denotes floor function. WFL
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