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

1,20

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

Table of n, a(n) for n=1..104.

Hans L. Bodlaender, The Chess Variant Pages

Fairbairn, Leggett et al., Information about Shogi (Japanese chess)

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 *)

Crossrefs

A009766, A049604, A062104, trinv given at A054425.

Sequence in context: A270599 A091398 A374175 * A112314 A350872 A280799

Adjacent sequences: A062100 A062101 A062102 * A062104 A062105 A062106

Keyword

nonn,tabl

Author

Antti Karttunen, May 30 2001

Status

approved