Hello Avni, I am surprised with the discrepancy you have found in the a(6) case. I still hold hope that your formula does give the correct values for all a(n). Could it be that the discrepancy is due to Matlab's floating point arithmetic?
Since you and Mark have both noted the effect of parity of n, I want to point out that I have reason to believe that what matters is not whether n is odd or even, but whether it is prime or composite. I don't have a fool-proof argument to support that right now; just a preliminary analysis. I will write about this again if I find out more.
---- original message ------------------------------------------- From: Avni Pllana <firstname.lastname@example.org> Date: Wed, 14 Jul 2010 12:14:51 +0000 (UTC) To: email@example.com To: firstname.lastname@example.org Subject: Re: Tiling the plane with checkerboard patterns
> 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 >
I tried to find further numerical evidence about my sum formula. A comparison of the number of equivalence classes using the Matlab program and using the sum formula is shown below:
Mat_1=[1 1] Sum_1=[1 1]
a(1)=2, Matlab a(1)=2, sum formula ________________________
Mat_2=[1 1 3 1 1] Sum_2=[1 1 3 1 1]
a(2)=7, Matlab a(2)=7, sum formula ________________________