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

0,3

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..69.

Hans L. Bodlaender, The Chess Variant Pages

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

Example

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,
...

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 *)

Prog

(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)))

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.

Sequence in context: A129322 A089984 A284429 * A210563 A236311 A347773

Adjacent sequences: A062102 A062103 A062104 * A062106 A062107 A062108

Keyword

nonn,tabl

Author

Antti Karttunen, May 30 2001

Extensions

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

Status

approved