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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. 4
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A254281 A295682 A195772 * 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

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Last modified December 12 06:05 EST 2017. Contains 295937 sequences.