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Topic: Tiling the plane with checkerboard patterns
Replies: 21   Last Post: Jul 14, 2010 10:33 PM

 Messages: [ Previous | Next ]
 Rouben Rostamian Posts: 193 Registered: 12/6/04
Re: Tiling the plane with checkerboard patterns
Posted: Jul 9, 2010 5:45 PM

On Fri, Jul 09, 2010 at 06:00:32PM +0000, Avni Pllana wrote:
>
> I think I found the right algorithm/formula for a(n):
>
> %%%%%%%%%%%%%%%%%%%%%%%%
>
> for n=1:5
>
> N=n^2;
> A=2;
>
> for i=1:N-1
> A=A+floor(nchoosek(N,i)/N)+mod(nchoosek(N,i),N);
> end
>
> a(n)=A;
>
> end
>
> %%%%%%%%%%%%%%%%%%%%%%%%

Avni, that's wonderful! It produces the exact
values for a(n), n=1,2,3,4,5. Do you have a proof
for the general case?

Rouben

Aside: Since the Matlab program above is limited
by its dependence on floating point arithmetic
to the range n=1,2,...,7, I recoded your algorithm
in Maple which performs integer arithmetic with no
limits. Here is the Maple's version of your formula:

a := proc(n::posint)
local N, b;
N := n^2;
b := binomial(N,i);
return sum( iquo(b,N) + irem(b, N), i=0..N);
end proc:

Then the command "seq(a(n), n=1..9);" produces
the values of a(n) for n=1,2,...,9:

n a(n)
1 2
2 7
3 64
4 4156
5 1342208
6 1908874521
7 11488774559744
8 288230376151712689
9 29850020237398251228192

Date Subject Author
7/6/10 Rouben Rostamian
7/7/10 Avni Pllana
7/8/10 mark
7/9/10 Rouben Rostamian
7/9/10 Avni Pllana
7/9/10 Rouben Rostamian
7/10/10 Avni Pllana
7/10/10 mark
7/11/10 Rouben Rostamian
7/11/10 mark
7/11/10 Rouben Rostamian
7/11/10 mark
7/12/10 Rouben Rostamian
7/13/10 mark
7/14/10 Rouben Rostamian
7/14/10 mark
7/11/10 Rouben Rostamian
7/14/10 Avni Pllana
7/14/10 Rouben Rostamian
7/13/10 Mary Krimmel
7/14/10 Rouben Rostamian
7/14/10 mark