[seqfan] israel at math.ubc.ca

[David Wilson]

On 6/24/2012 9:33 PM, israel at math.ubc.ca wrote:
> The formula seems to be correct. It arises as follows. Let A be the 64 
> x 64 adjacency matrix of a chessboard, with edges corresponding to 
> king-moves. Then A025597(n) is the matrix element of A^n corresponding 
> to two opposite corners of the board, and thus a certain linear 
> combination of the n'th powers of the eigenvalues of A. Now A + I = (B 
> + I) tensor (B + I) where B is the adjacency matrix of [1,2,...,8] 
> with nearest-neighbour edges. The characteristic polynomial of B is 
> t^8-7*t^6+15*t^4-10*t^2+1, which is U_8(t/2) where U_8 is the 8'th 
> Chebyshev polynomial of the second kind.
>
> Robert Israel
> University of British Columbia

Clearly I'm not at your level of mathematical understanding.

The Chebyshev form of this polynomial implies something nice about the 
eigenvalues?