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A062103
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Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.
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4
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0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 14
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OFFSET
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1,20
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COMMENTS
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Table formatted as a square array shows the top-left corner of the infinite board. This is an aerated and sligthly skewed variant of Catalan's triangle A009766.
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LINKS
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MAPLE
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[seq(ShoogiKnightSeq(j), j=1..120)]; ShoogiKnightSeq := n -> ShoogiKnightTriangle(trinv(n-1)-1, (n-((trinv(n-1)*(trinv(n-1)-1))/2))-1);
ShoogiKnightTriangle := proc(r, m) option remember; if(m < 0) then RETURN(0); fi; if(r < 0) then RETURN(0); fi; if(m > r) then RETURN(0); fi; if((1 = r) and (0 = m)) then RETURN(1); fi; RETURN(ShoogiKnightTriangle(r-3, m-2) + ShoogiKnightTriangle(r-1, m-2)); end;
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MATHEMATICA
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trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
ShoogiKnightSeq[n_] := ShoogiKnightTriangle[trinv[n - 1] - 1, (n - ((trinv[n - 1]*(trinv[n - 1] - 1))/2)) - 1];
ShoogiKnightTriangle[r_, m_] := ShoogiKnightTriangle[r, m] = Which[m < 0, 0, r < 0, 0, m > r, 0, r == 1 && m == 0, 1, True, ShoogiKnightTriangle[r - 3, m - 2] + ShoogiKnightTriangle[r - 1, m - 2]];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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