
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:N1 > 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

