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.

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, 1, 3, 9, 27, 80, 216, 483, 750, 1, 3, 9, 27, 81, 236, 623, 1373, 2123, 1, 3, 9, 27, 81, 242, 694, 1800, 3923, 6046, 1, 3, 9, 27, 81, 243, 721, 2038, 5211, 11257, 17303, 1, 3, 9, 27

Offset: 0

Author

Antti Karttunen, May 30 2001

Keywords

• nonn
• tabl

Comments

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

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

Examples

			Array begins:
 1       1       1       1       1       1       1       1       1       1       1
 2       3       3       3       3       3       3       3       3       3       3
 5       8       9       9       9       9       9       9       9       9 ...
 13      22      26      27      27      27      27      27      27 ...
 35      61      75      80      81      81      81 ...
 96      171     216     236     242     243 ...
 267     483     623     694     721 ...
 750     1373    1800    2038 ...
 2123    3923    5211 ...
 6046    11257 ...
 17303  ...
 ...
Formatted as a triangle:
 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,
 1, 3, 9, 27, 80, 216, 483, 750,
 1, 3, 9, 27, 81, 236, 623, 1373, 2123,
 1, 3, 9, 27, 81, 242, 694, 1800, 3923, 6046,
 1, 3, 9, 27, 81, 243, 721, 2038, 5211, 11257, 17303,
 ...
		

Links

• Hans L. Bodlaender, The Chess Variant Pages
• Hans L. Bodlaender et al., editors, The Piececlopedia (An overview of several fairy chess pieces)

Crossrefs

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.

Programs

Maple

[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) );
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;
M:=12; T:=Array(0..M,0..M,0);
T[0,0]:=1; T[1,1]:=1;
for i from 1 to M do T[i,0]:=0; od:
for n from 2 to M do for k from 1 to n do
T[n,k]:= T[n,k-1]+T[n-1,k-1]+T[n-2,k-1];
od: od;
rh:=n->[seq(T[n,k],k=0..n)];
for n from 0 to M do lprint(rh(n)); od: # N. J. A. Sloane, Apr 11 2020

Mathematica

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 *)

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)))

Extensions

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