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A272469 Numbers of n-step paths of a king moving on an n X n chessboard, starting at a corner and not visiting any cell twice. 2
0, 6, 50, 322, 1874, 10558, 58716, 325758, 1808778, 10068548, 56213606, 314785072, 1767660604, 9951449844, 56151698716, 317484868212, 1798343124800, 10203031413894 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..18.

EXAMPLE

On an n X n chessboard, a king in a corner is allowed to have n moves. For n=2, let's name the cells A1,A2,B1,B2 with the king at A1. Two moves, without repeating cells, can be done in the following 6 different ways: {A1-A2-B1, A1-A2-B2, A1-B1-A2, A1-B1-B2, A1-B2-A2, A1-B2-B1}. So a(2)=6.

MAPLE

pathCount := proc (N)

local g1, g2, nStep, gg, nCells, nPrev, i1, i2, j1, j2, i, j, nNext;

    nCells := N^2; g1 := [[1]];

    if N = 1 then return nops(g1) fi; #forced value for N=0

    for nStep to N do

        g2 := [];

        for gg in g1 do

            nPrev := gg[-1];

            i1 := `if`(floor((nPrev-1)/N) = 0, 0, -N);

            i2 := `if`(floor((nPrev-1)/N) = N-1, 0, N);

            j1 := `if`(`mod`(nPrev-1, N) = 0, 0, -1);

            j2 := `if`(`mod`(nPrev-1, N) = N-1, 0, 1);

            for i from i1 by N to i2 do

                for j from j1 to j2 do

                    if i = 0 and j = 0 then next fi;

                    nNext := nPrev+i+j;

                    if nNext < 0 or nCells < nNext or (nNext in gg) then next fi;

                    g2 := [op(g2), [op(gg), nNext]]

                end do

            end do

        end do;

        g1 := g2

    end do;

    return nops(g1);

end proc:

[seq(pathCount(n), n = 1 .. 6)];

CROSSREFS

Cf. A272445.

Sequence in context: A220887 A213807 A241781 * A223816 A180880 A308860

Adjacent sequences:  A272466 A272467 A272468 * A272470 A272471 A272472

KEYWORD

nonn,walk,more

AUTHOR

César Eliud Lozada, Apr 30 2016

EXTENSIONS

a(9)-a(16) from Alois P. Heinz, May 01 2016

a(17)-a(18) from Bert Dobbelaere, Jan 08 2019

STATUS

approved

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Last modified August 18 13:52 EDT 2019. Contains 326100 sequences. (Running on oeis4.)