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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified November 26 21:07 EST 2021. Contains 349344 sequences. (Running on oeis4.)