A062104 -id:A062104 - cp's OEIS frontend

A062107 Diagonal of table A062104.

0, 1, 3, 10, 30, 90, 270, 810, 2430, 7290, 21870, 65610, 196830, 590490, 1771470, 5314410, 15943230, 47829690, 143489070, 430467210, 1291401630, 3874204890, 11622614670, 34867844010, 104603532030, 313810596090, 941431788270

Offset: 1

Antti Karttunen, May 30 2001

nonn

Index entries for linear recurrences with constant coefficients, signature (3).

Except for initial terms, same as A005052.

Maple
```
[seq(ChessPawnTriangle(j,j),j=1..50)];
```

a(n) = 10*3^(n-4) for n >= 4.
From Paul Barry, Oct 15 2004: (Start)
G.f.: x^2(1+x^2)/(1-3x);
a(n) = Sum_{k=0..n-2} 3^(n-k-2)binomial(1, k/2)(1+(-1)^k)/2. (End)

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.

Original entry on oeis.org

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

Antti Karttunen, May 30 2001

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

			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,
 ...

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

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.

[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

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 *)

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)))

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

A062103 Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.

Original entry on oeis.org

0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 14

Offset: 1

Antti Karttunen, May 30 2001

Table formatted as a square array shows the top-left corner of the infinite board. This is an aerated and sligthly skewed variant of Catalan's triangle A009766.

Hans L. Bodlaender, The Chess Variant Pages
Fairbairn, Leggett et al., Information about Shogi (Japanese chess)

A009766, A049604, A062104, trinv given at A054425.

[seq(ShoogiKnightSeq(j),j=1..120)]; ShoogiKnightSeq := n -> ShoogiKnightTriangle(trinv(n-1)-1,(n-((trinv(n-1)*(trinv(n-1)-1))/2))-1);
ShoogiKnightTriangle := proc(r,m) option remember; if(m < 0) then RETURN(0); fi; if(r < 0) then RETURN(0); fi; if(m > r) then RETURN(0); fi; if((1 = r) and (0 = m)) then RETURN(1); fi; RETURN(ShoogiKnightTriangle(r-3,m-2) + ShoogiKnightTriangle(r-1,m-2)); end;

trinv[n_] := Floor[(1 + Sqrt[8 n + 1])/2];
ShoogiKnightSeq[n_] := ShoogiKnightTriangle[trinv[n - 1] - 1, (n - ((trinv[n - 1]*(trinv[n - 1] - 1))/2)) - 1];
ShoogiKnightTriangle[r_, m_] := ShoogiKnightTriangle[r, m] = Which[m < 0, 0, r < 0, 0, m > r, 0, r == 1 && m == 0, 1, True, ShoogiKnightTriangle[r - 3, m - 2] + ShoogiKnightTriangle[r - 1, m - 2]];
Array[ShoogiKnightSeq, 120] (* Jean-François Alcover, Mar 06 2016, adapted from Maple *)

A062106 Number of ways a black pawn (from any starting square on the second back rank) can (theoretically) end on the n-th square of the leftmost file counted from the back rank.

Original entry on oeis.org

0, 1, 2, 6, 15, 40, 109, 302, 846, 2390, 6796, 19426, 55767, 160668, 464305, 1345282, 3906701, 11367696, 33135987, 96740610, 282831981, 827939880, 2426431239, 7118546874, 20904025380, 61439768166, 180725813478, 532004277518

Offset: 1

Antti Karttunen, May 30 2001

nonn

The left column of table A062104. Cf. also A062107.

Maple
```
[seq(ChessPawnTriangle(j,1),j=1..40)];
```

cp's OEIS Frontend

A062107 Diagonal of table A062104.

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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.

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Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A062103 Number of paths by which an unpromoted knight (keima) of Shogi can move to various squares on infinite board, if it starts from its origin square, the second leftmost square of the back rank.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A062106 Number of ways a black pawn (from any starting square on the second back rank) can (theoretically) end on the n-th square of the leftmost file counted from the back rank.

Views

Author

Keywords

Crossrefs

Programs

Maple