[seqfan] Re: A025597 computation

[israel at math.ubc.ca]

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

On Jun 24 2012, Wouter Meeussen wrote:

>yes, the result fits all 23 known terms; extension to 48 :
>
>Table[4/81*
>      Sum[(-1)^(j + k)  *
>          Sin[j*Pi/9]^2  Sin[
>              k*Pi/9]^2 *  ((1 + 2Cos[j*Pi/9])*(1 + 2Cos[k*Pi/9]) - 1)^n , 
>{j,
>           8}, {k, 8}], {n, 48}];
>
>Round[ N[%, 64] ]
>
> {0, 0, 0, 0, 0, 0, 1, 56, 1309, 20370, 255366, 2782296, 27630317, 
> 256617790, 2269878170, 19345170656, 160223380546, 1297456951652, 
> 10319966008680, 80906898257760, 626886465395595, 4810654849509082, 
> 36623649326935517, 276978367797824968, 2083200939963715126, 
> 15595683813101279420, 116301107197615295905, 864435471139653592500, 
> 6407217503910893277444, 47378067330093217182600, 
> 349630722268949591264588, 2575700394882062294904448, 
> 18947140880906832557051456, 139202483079923884540608128, 
> 1021606539041179715511956893, 7490666332544968034191929020, 
> 54880014410659600034779737717, 401803308907807919979966340230, 
> 2940084653973218884807223098446, 21502488469368326228248902945424, 
> 157192063747654277121963377183669, 1148713224897468635134162828539394, 
> 8391781153727818906899077746007038, 61288282367565060553519666623733112, 
> 447505180064735246955442905726814174, 
> 3266857758785435955732406250575642164, 
> 23844387150444635926903104779107943592, 
> 174010885751792255799158735894761920384}
>
>Wouter
>
>
>-----Original Message----- 
>From: David Wilson
>Sent: Sunday, June 24, 2012 4:19 PM
>To: Sequence Fanatics
>Subject: [seqfan] A025597 computation
>
>I have been sent the following ostensible formula for A025597.
>*
>f(n) = (4/81) * sum(j=1...8, k=1...8) [(-1)^(j+k)] *
>[sin(j*pi/9)*sin(k*pi/9)]^2 * [(1+2cos(j*pi/9))*(1+2cos(k*pi/9))-1]^n
>
>*Could someone with power tools kindly verify a few terms?
>*
>*
>
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