%I #6 Mar 06 2016 03:23:32
%S 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,
%T 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,
%U 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
%N 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.
%C 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.
%H Hans L. Bodlaender, <a href="http://www.chessvariants.com/index.html">The Chess Variant Pages</a>
%H Fairbairn, Leggett et al., <a href="http://www.iki.fi/~kartturi/shogi/shoogi.htm">Information about Shogi (Japanese chess)</a>
%p [seq(ShoogiKnightSeq(j),j=1..120)]; ShoogiKnightSeq := n -> ShoogiKnightTriangle(trinv(n-1)-1,(n-((trinv(n-1)*(trinv(n-1)-1))/2))-1);
%p 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;
%t trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
%t ShoogiKnightSeq[n_] := ShoogiKnightTriangle[trinv[n - 1] - 1, (n - ((trinv[n - 1]*(trinv[n - 1] - 1))/2)) - 1];
%t 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]];
%t Array[ShoogiKnightSeq, 120] (* _Jean-François Alcover_, Mar 06 2016, adapted from Maple *)
%Y A009766, A049604, A062104, trinv given at A054425.
%K nonn,tabl
%O 1,20
%A _Antti Karttunen_, May 30 2001