
Re: Tiling the plane with checkerboard patterns
Posted:
Jul 11, 2010 1:58 PM


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)(n1)(n+1))/n^2 > > My formula for even N is worse. It already disagrees at N=6 > > A(N_even)= ((2^(n^2)+ (2^(2n2))(n1)(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 nonintegers 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)(n1)(n+1)), n^2) A(N_even) = Quo( (2^(n^2)+ (2^(2n2))(n1)(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

