[seqfan] Re: n x n matrices with adjacent entries differing by +/- 1

Edwin Clark eclark at math.usf.edu
Sun Nov 2 02:54:37 CET 2008

On Mon, 27 Oct 2008, Benoît Jubin wrote:
> It would also be interesting to consider the entries of the matrix in
> Z/kZ (that is, 1 and k would also differ by 1). And also the same
> sequence for entries in Z or N, the upper-left term being 0.

I computed for n = 1 to 8 the number of n x n matrices with entries in Z, 
the upper left entry = 0, and adjacent entries (in the same row or column) 
differing by +/- 1. I got the following:

1, 6, 82, 2604, 193662, 33865632, 13956665236, 13574876544396

This  matches the first 8 terms of


which is defined as:

  1/3 of the number of colorings of an n X n square array with 3 colors

It is too much to imagine the sequences are different, yet I also cannot 
imagine they coincide by looking at the definitions. Does anyone see a 


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