cp's OEIS Frontend

A062105 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.

%I A062105 #39 Sep 20 2024 06:14:09
%S A062105 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 A062105 1,3,9,27,80,216,483,750,1,3,9,27,81,236,623,1373,2123,1,3,9,27,81,
%U A062105 242,694,1800,3923,6046,1,3,9,27,81,243,721,2038,5211,11257,17303,1,3,9,27
%N A062105 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 A062105 Table formatted as a square array shows the top-left corner of the infinite board.
%C A062105 The same array can also be constructed by the method used for constructing A217536, except with a top row consisting entirely of 1's instead of the natural numbers. - _WG Zeist_, Aug 25 2024
%H A062105 Hans L. Bodlaender, <a href="http://www.chessvariants.com/index.html">The Chess Variant Pages</a>
%H A062105 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 A062105 Array begins:
%e A062105  1       1       1       1       1       1       1       1       1       1       1
%e A062105  2       3       3       3       3       3       3       3       3       3       3
%e A062105  5       8       9       9       9       9       9       9       9       9 ...
%e A062105  13      22      26      27      27      27      27      27      27 ...
%e A062105  35      61      75      80      81      81      81 ...
%e A062105  96      171     216     236     242     243 ...
%e A062105  267     483     623     694     721 ...
%e A062105  750     1373    1800    2038 ...
%e A062105  2123    3923    5211 ...
%e A062105  6046    11257 ...
%e A062105  17303  ...
%e A062105  ...
%e A062105 Formatted as a triangle:
%e A062105  1,
%e A062105  1, 2,
%e A062105  1, 3, 5,
%e A062105  1, 3, 8, 13,
%e A062105  1, 3, 9, 22, 35,
%e A062105  1, 3, 9, 26, 61, 96,
%e A062105  1, 3, 9, 27, 75, 171, 267,
%e A062105  1, 3, 9, 27, 80, 216, 483, 750,
%e A062105  1, 3, 9, 27, 81, 236, 623, 1373, 2123,
%e A062105  1, 3, 9, 27, 81, 242, 694, 1800, 3923, 6046,
%e A062105  1, 3, 9, 27, 81, 243, 721, 2038, 5211, 11257, 17303,
%e A062105  ...
%p A062105 [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 A062105 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 A062105 M:=12; T:=Array(0..M,0..M,0);
%p A062105 T[0,0]:=1; T[1,1]:=1;
%p A062105 for i from 1 to M do T[i,0]:=0; od:
%p A062105 for n from 2 to M do for k from 1 to n do
%p A062105 T[n,k]:= T[n,k-1]+T[n-1,k-1]+T[n-2,k-1];
%p A062105 od: od;
%p A062105 rh:=n->[seq(T[n,k],k=0..n)];
%p A062105 for n from 0 to M do lprint(rh(n)); od: # _N. J. A. Sloane_, Apr 11 2020
%t A062105 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 A062105 (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 A062105 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, A217536.
%K A062105 nonn,tabl
%O A062105 0,3
%A A062105 _Antti Karttunen_, May 30 2001
%E A062105 Edited by _N. J. A. Sloane_, May 22 2014