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 A025597 Number of n-move king paths on 8 X 8 board from given corner to opposite corner. 1
 0, 0, 0, 0, 0, 0, 0, 1, 56, 1309, 20370, 255366, 2782296, 27630317, 256617790, 2269878170, 19345170656, 160223380546, 1297456951652, 10319966008680, 80906898257760, 626886465395595, 4810654849509082, 36623649326935517 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS As long as the path starts and ends on the correct square, it is counted. The path may revisit squares, including the start and end squares. LINKS David W. Wilson and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 49 terms from David W. Wilson) FORMULA a(n) = (4/81) * Sum_{j,k=1..8} (-1)^(j+k)* [sin(j*Pi/9)*sin(k*Pi/9)]^2 *[(1+2*cos(j*Pi/9))*(1+2*cos(k*Pi/9))-1]^n. - Andrew G. Buchanan, Jun 24 2012 G.f.: -(408459*x^21 +3108249*x^20 +5135985*x^19 -8733022*x^18 -29723403*x^17 -6771900*x^16 +52706117*x^15 +58351590*x^14 -6069219*x^13 -51965240*x^12 -37661505*x^11 -6328524*x^10 +5718540*x^9 +3500727*x^8 +471552*x^7 -208258*x^6 -90243*x^5 -9609*x^4 +1531*x^3 +498*x^2 +42*x+1) *x^7 / ((3*x-1) *(x+1) *(3*x^3-3*x-1) *(x^3-3*x+1) *(17*x^3+6*x^2-3*x-1) *(x^3+3*x^2-6*x+1) *(3*x^3+9*x^2+6*x-1) *(19*x^3-9*x^2-3*x+1) *(x^3+9*x^2+6*x+1) *(3*x^3-9*x^2-3*x+1) *(17*x^3+18*x^2+3*x-1)). - Alois P. Heinz, Jun 25 2012 EXAMPLE The king cannot reach the opposite corner in fewer than 7 moves, hence a(0) through a(6) are all 0. There is only one way to reach the opposite corner in 7 moves, namely along the main diagonal, so a(7) = 1. MAPLE b:= proc(n, i, j) option remember;       `if`(n<0 or i<0 or i>7 or j<0 or j>7, 0, `if`([n, i, j]=[0\$3],        1, add(b(n-1, i+r[1], j+r[2]), r=[[1, 1], [1, 0], [1, -1],           [0, 1], [0, -1], [-1, 1], [-1, 0], [-1, -1]])))     end: a:= n-> b(n, 7, 7): seq(a(n), n=0..30);  # Alois P. Heinz, Jun 25 2012 MATHEMATICA f[n_] := Round[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}]]; Array[f, 23] (* Robert G. Wilson v, Jun 28 2012 *) b[n_, i_, j_] := b[n, i, j] = If[n<0 || i<0 || i>7 || j<0 || j>7, 0, If[{n, i, j} == {0, 0, 0}, 1, Sum [b[n-1, i+r[[1]], j+r[[2]]], {r, {{1, 1}, {1, 0}, {1, -1}, {0, 1}, {0, -1}, {-1, 1}, {-1, 0}, {-1, -1}}}]]]; a[n_] := b[n, 7, 7]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *) CROSSREFS Sequence in context: A183587 A219469 A278079 * A034202 A160290 A030649 Adjacent sequences:  A025594 A025595 A025596 * A025598 A025599 A025600 KEYWORD nonn,easy AUTHOR EXTENSIONS a(24)-a(48) from Wouter Meeussen, Jun 24 2012 STATUS approved

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