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Re: Tiling the plane with checkerboard patterns
Posted:
Jul 11, 2010 1:58 PM
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On Sun, Jul 11, 2010 at 04:02:08AM +0000, mark wrote:
>
> I just checked my formula against Avni's extended list.
> We are very close. My odd N formula agrees at N=7
> and disagrees slightly at N=9.
>
> A(N_odd)= ((2^(n^2)+ (2^n)(n-1)(n+1))/n^2
>
> My formula for even N is worse. It already disagrees at N=6
>
> A(N_even)= ((2^(n^2)+ (2^(2n-2))(n-1)(n+1))/n^2
>
> Note: If this chart does not display properly, viewing
> as text might help.
>
> > n a(n,Avni) a(n,Mark)
> > 1 2 2
> > 2 7 7
> > 3 64 64
> > 4 4156 4156
> > 5 1342208 1342208
> > 6 1908874521 1908875349 1/3
> > 7 11488774559744 11488774559744
> > 8 288230376151712689 288230376151727872
> > 9 29850020237398251228192 29850020237398251227818 2/3
That's a surprisingly good fit but it's unfortunate that the
results are non-integers for n=6 and n=9. Avni's formula is
more credible since it is guaranteed to produce integers.
On the second thought, you can force integer results from your
formulas by replacing your .../n^2 by Quo(...,n^2) where
where Quo(p,q) is the quotient of the integers p and q. Thus:
A(N_odd) = Quo( (2^(n^2)+ (2^n)(n-1)(n+1)), n^2)
A(N_even) = Quo( (2^(n^2)+ (2^(2n-2))(n-1)(n+1)), n^2)
Then you will get:
n a(n,Avni) a(n,Mark,modified)
1 2 2
2 7 7
3 64 64
4 4156 4156
5 1342208 1342208
6 1908874521 1908875349
7 11488774559744 11488774559744
8 288230376151712689 288230376151727872
9 29850020237398251228192 29850020237398251227818
I don't know which one is closer to the truth.
Rouben
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