From: Csaba Henk <csaba@..._for_avoiding_spam.org>
Date: 2005-04-11T17:04:36+09:00
Subject: Re: Translating A Pattern of Data Into Equation, and ultimately code

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