A062103 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.

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

Offset: 1

Author

Antti Karttunen, May 30 2001

Keywords

• nonn
• tabl

Comments

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.

Links

• Hans L. Bodlaender, The Chess Variant Pages
• Fairbairn, Leggett et al., Information about Shogi (Japanese chess)

Crossrefs

A009766, A049604, A062104, trinv given at A054425.

Programs

Maple

[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;

Mathematica

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]];
Array[ShoogiKnightSeq, 120] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)