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%I #30 Apr 11 2020 15:00:10
%S 1,1,2,1,3,5,1,3,8,13,1,3,9,22,35,1,3,9,26,61,96,1,3,9,27,75,171,267,
%T 1,3,9,27,80,216,483,750,1,3,9,27,81,236,623,1373,2123,1,3,9,27,81,
%U 242,694,1800,3923,6046,1,3,9,27,81,243,721,2038,5211,11257,17303,1,3,9,27
%N Square array read by antidiagonals: number of ways a pawn-like piece (with the initial 2-step move forbidden and starting from any square on the back rank) can end at various squares on an infinite chessboard.
%C Table formatted as a square array shows the top-left corner of the infinite board.
%H Hans L. Bodlaender, <a href="http://www.chessvariants.com/index.html">The Chess Variant Pages</a>
%H Hans L. Bodlaender et al., editors, <a href="http://www.chessvariants.com/piececlopedia.dir/index.html">The Piececlopedia (An overview of several fairy chess pieces)</a>
%e Array begins:
%e 1 1 1 1 1 1 1 1 1 1 1
%e 2 3 3 3 3 3 3 3 3 3 3
%e 5 8 9 9 9 9 9 9 9 9 ...
%e 13 22 26 27 27 27 27 27 27 ...
%e 35 61 75 80 81 81 81 ...
%e 96 171 216 236 242 243 ...
%e 267 483 623 694 721 ...
%e 750 1373 1800 2038 ...
%e 2123 3923 5211 ...
%e 6046 11257 ...
%e 17303 ...
%e ...
%e Formatted as a triangle:
%e 1,
%e 1, 2,
%e 1, 3, 5,
%e 1, 3, 8, 13,
%e 1, 3, 9, 22, 35,
%e 1, 3, 9, 26, 61, 96,
%e 1, 3, 9, 27, 75, 171, 267,
%e 1, 3, 9, 27, 80, 216, 483, 750,
%e 1, 3, 9, 27, 81, 236, 623, 1373, 2123,
%e 1, 3, 9, 27, 81, 242, 694, 1800, 3923, 6046,
%e 1, 3, 9, 27, 81, 243, 721, 2038, 5211, 11257, 17303,
%e ...
%p [seq(CPTVSeq(j),j=0..91)]; CPTVSeq := n -> ChessPawnTriangleV( (2+(n-((trinv(n)*(trinv(n)-1))/2))), ((((trinv(n)-1)*(((1/2)*trinv(n))+1))-n)+1) );
%p ChessPawnTriangleV := 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; RETURN(ChessPawnTriangleV(r-1,c-1)+ChessPawnTriangleV(r-1,c)+ChessPawnTriangleV(r-1,c+1)); end;
%p M:=12; T:=Array(0..M,0..M,0);
%p T[0,0]:=1; T[1,1]:=1;
%p for i from 1 to M do T[i,0]:=0; od:
%p for n from 2 to M do for k from 1 to n do
%p T[n,k]:= T[n,k-1]+T[n-1,k-1]+T[n-2,k-1];
%p od: od;
%p rh:=n->[seq(T[n,k],k=0..n)];
%p for n from 0 to M do lprint(rh(n)); od: # _N. J. A. Sloane_, Apr 11 2020
%t T[n_, k_] := T[n, k] = If[n < 1 || k < 1, 0, If[n == 1, 1, T[n - 1, k - 1] + T[n - 1, k] + T[n - 1, k + 1]]]; Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Mar 04 2016, adapted from PARI *)
%o (PARI) T(n,k)=if(n<1 || k<1,0,if(n==1,1,T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)))
%Y A005773 gives the left column of the table. A000244 (powers of 3) gives the diagonal of the table. Variant of A062104. Cf. also A062103, A020474.
%K nonn,tabl
%O 0,3
%A _Antti Karttunen_, May 30 2001
%E Edited by _N. J. A. Sloane_, May 22 2014
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