A025595 Number of n-move king paths on 8 X 8 board from given corner to any square.

1, 3, 18, 105, 684, 4550, 31340, 219555, 1559835, 11177190, 80573373, 583082082, 4230475092, 30745373256, 223692489189, 1628685532071, 11863934648190, 86447927584521, 630040117048380, 4592393936655066, 33477073967039868, 244050825770095095, 1779217416632500479

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Ph. Flajolet, March 28 2013.

Natalia L. Skirrow, chessboards (section 'the fourierful king of chebyshev')

Index entries for linear recurrences with constant coefficients, signature (9,9,-159,-108,810,900,-513,-729,-27,81).

Formula

G.f.: -(48*x^9 +120*x^8 -126*x^7 -457*x^6 -175*x^5 +162*x^4 +75*x^3 -18*x^2 -6*x+1) / ((3*x-1) *(3*x^3-3*x-1) *(3*x^3-9*x^2-3*x+1) *(3*x^3+9*x^2+6*x-1)). - Alois P. Heinz, Jun 25 2012

a(n) = (2/9)^2 * Sum_{i,j=1..7, i,j odd} (1+cos(i*Pi/9)) * (1+cos(j*Pi/9)) * ((1+2*cos(i*Pi/9)) * (1+2*cos(j*Pi/9)) - 1)^n. - Natalia L. Skirrow, Feb 20 2026

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=0,
       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, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Jun 25 2012

Mathematica

b[n_, i_, j_] := b[n, i, j] = If[n<0 || i<0 || i>7 || j<0 || j>7, 0, If[n == 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, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2015, after Alois P. Heinz *)

Crossrefs

Sequence in context: A106328 A391219 A007277 * A151331 A137962 A267662

Adjacent sequences: A025592 A025593 A025594 * A025596 A025597 A025598

Keyword

nonn,easy

Author

David W. Wilson

Status

approved