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%I #22 Mar 06 2016 03:24:00
%S 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,
%T 3,10,30,84,193,302,0,1,3,10,30,89,242,544,846,0,1,3,10,30,90,263,698,
%U 1544,2390,0,1,3,10,30,90,269,774,2016,4406,6796,0,1,3,10,30,90,270
%N 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.
%C Table formatted as a square array shows the top-left corner of the infinite board.
%e Array begins:
%e 0 0 0 0 0 0 0 0 0 0 0 0 ...
%e 1 1 1 1 1 1 1 1 1 1 1 ...
%e 2 3 3 3 3 3 3 3 3 3 ...
%e 6 9 10 10 10 10 10 10 10 ...
%e 15 25 29 30 30 30 30 30 ...
%e 40 69 84 89 90 90 90 ...
%e 109 193 242 263 269 270 ...
%e 302 544 698 774 ...
%e 846 1544 2016 ...
%e 2390 4406 ...
%e 6796 ...
%p [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) );
%p 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));
%p else RETURN(ChessPawnTriangle(r-1,c-1)+ChessPawnTriangle(r-1,c)+ChessPawnTriangle(r-1,c+1)); fi; end;
%t trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
%t CPTSeq[n_] := ChessPawnTriangle[(1 + (n - ((trinv[n]*(trinv[n] - 1))/2))), ((((trinv[n] - 1)*(((1/2)*trinv[n]) + 1)) - n) + 1)];
%t 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]];
%t Table[CPTSeq[j], {j, 0, 91}] (* _Jean-François Alcover_, Mar 06 2016, adapted from Maple *)
%Y 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.
%K nonn,tabl
%O 0,6
%A _Antti Karttunen_, May 30 2001
%E Edited by _N. J. A. Sloane_, May 22 2014
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