Csaba Henk
4/11/2005 7:58:00 AM
On 2005-04-11, Edgardo Hames <ehames@gmail.com> wrote:
> On Apr 10, 2005 8:45 PM, Zach Dennis <zdennis@mktec.com> wrote:
> For columns numbered with a power of two (1, 2, 4, 8, 16, ...) the deltas are
> the power of two alternating its sign +/-.
> It's rather difficult to explain, so please, take a look at the following table:
>
> Input 1 2 3 4 5 6 7 8 9 10 11 12
> -----------------------------------
> Col 1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1 -1 +1
> Col 2 +2 -2 -2 +2 +2 -2 -2 +2 -2 -2 +2 +2
> Col 4 +4 +4 +4 -4 -4 -4 -4 +4 +4 +4 +4 -4
> Col 8 +8 +8 +8 +8 +8 +8 +8 -8 -8 -8 -8 -8
That's sort of cool! Congrats!
I don't think there would be a concise expression for this table. I
mean, there is no two-variable polynomial which would describe this
table.
Proof: assume there is p(x,y) s.t. the i-th element of the j-th row can
be obtained as p(i,j). Then fix j=1. Thus you get a polynomial q(x) as
well. Take the delta of it: d(x) = q(x) - q(x-1). That's a polynomial
again, and we know its values at positive integers: d(n) = 1 if n is
even and d(n) = -1 if n is odd. As you'd write in ruby, d(n).abs <= 1.
But that's a contradiction, as polynomials tend to either plus or minus
infinity at infinity.
So the best formalization is writing an algorithm which calculates table
values based on the description Edgardo gave.
Csaba