A062104 Square array read by antidiagonals: number of ways a black pawn (starting at any square on the second rank) can (theoretically) end at various squares on an infinite chessboard.

0, 0, 1, 0, 1, 2, 0, 1, 3, 6, 0, 1, 3, 9, 15, 0, 1, 3, 10, 25, 40, 0, 1, 3, 10, 29, 69, 109, 0, 1, 3, 10, 30, 84, 193, 302, 0, 1, 3, 10, 30, 89, 242, 544, 846, 0, 1, 3, 10, 30, 90, 263, 698, 1544, 2390, 0, 1, 3, 10, 30, 90, 269, 774, 2016, 4406, 6796, 0, 1, 3, 10, 30, 90, 270

Offset

0,6

Comments

Table formatted as a square array shows the top-left corner of the infinite board.

Links

Table of n, a(n) for n=0..72.

Example

Array begins:
0       0       0       0       0       0       0       0       0       0       0       0 ...
1       1       1       1       1       1       1       1       1       1       1 ...
2       3       3       3       3       3       3       3       3       3 ...
6       9       10      10      10      10      10      10      10 ...
15      25      29      30      30      30      30      30 ...
40      69      84      89      90      90      90 ...
109     193     242     263     269     270 ...
302     544     698     774 ...
846     1544    2016 ...
2390    4406 ...
6796 ...

Maple

[seq(CPTSeq(j), j=0..91)]; CPTSeq := n -> ChessPawnTriangle( (1+(n-((trinv(n)*(trinv(n)-1))/2))), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) );
ChessPawnTriangle := proc(r, c) option remember; if(r < 2) then RETURN(0); fi; if(c < 1) then RETURN(0); fi; if(2 = r) then RETURN(1); fi; if(4 = r) then RETURN(1+ChessPawnTriangle(r-1, c-1)+ChessPawnTriangle(r-1, c)+ChessPawnTriangle(r-1, c+1));
else RETURN(ChessPawnTriangle(r-1, c-1)+ChessPawnTriangle(r-1, c)+ChessPawnTriangle(r-1, c+1)); fi; end;

Mathematica

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
CPTSeq[n_] := ChessPawnTriangle[(1 + (n - ((trinv[n]*(trinv[n] - 1))/2))), ((((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1)];
ChessPawnTriangle[r_, c_] := ChessPawnTriangle[r, c] = Which[r < 2, 0, c < 1, 0, 2 == r, 1, 4 == r, 1 + ChessPawnTriangle[r - 1, c - 1] + ChessPawnTriangle[r - 1, c] + ChessPawnTriangle[r - 1, c + 1], True, ChessPawnTriangle[r - 1, c - 1] + ChessPawnTriangle[r - 1, c] + ChessPawnTriangle[r - 1, c + 1]];
Table[CPTSeq[j], {j, 0, 91}] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

Crossrefs

A062106 gives the left column and A062107 the diagonal of the table. A062105 is a more regular variant. Cf. also A062103. Trinv given at A054425.

Sequence in context: A295682 A195772 A330618 * A257783 A226874 A267901

Adjacent sequences: A062101 A062102 A062103 * A062105 A062106 A062107

Keyword

nonn,tabl

Author

Antti Karttunen, May 30 2001

Extensions

Edited by N. J. A. Sloane, May 22 2014

Status

approved