Fred, OEIS led me to the following recent paper of Kitaev and Remmel which looks relevant. I haven't had a chance to look at it yet. Victor http://arxiv.org/abs/1304.4286 PS. Sorry about the terse replies. I had an accident two weeks ago which damaged my shoulder and makes it hard for me to type very much. On Sun, Feb 23, 2014 at 12:05 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Victor,
I'm unsure whether you meant this for the math-fun list? But anyway ...
Your A133130 problem is indeed intimately similar to Hardin's, and I'm most interested to know more about how far you got with it: presumably you actually had to involve more general m x n rectangles?
Do your recurrence polynomials have any striking properties? One instance: an extraordinary feature of the associated (monic) Hardin polynomial concerns its constant coefficient for m odd, which apparently equals (+/-) 2^k , where k equals the degree of the polynomial at m-1 ; while the constant coefficient of the bordered variant equals (+/-) 1 . [ m = 5 earlier has final coefficient -8 , while m = 4 has a cubic polynomial.]
Please tell a little more about the intriguing relation you mentioned between the fractal dimension d of the adjacency matrix, and the rate of growth c^(m n) of the counting function. For the Hardin case, I find that
d = Log(3)/Log(2) = 1.5849625 ...
which does indeed look as if it might act as an upper bound on c ~ 1.54 ; but the mechanism behind such a connection remains a mystery to me.
Fred Lunnon
On 2/23/14, Victor Miller <victorsmiller@gmail.com> wrote:
Fred, very nice stuff. This reminds me o something that I worked on many years ago briefly discussed in A133130: Number of 0/1 colorings of an n X n square for which no 2 by 2 subsquare is monochromatic. I didn't mention it in the comments, but the matrices involved had a nice recursive structure which,upon proper scaling, converged to a fractal whose Hausdorff dimension was related to the rate of growth.
Victor
On Sat, Feb 22, 2014 at 8:10 PM, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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