A217536 -id:A217536 - 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.

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

A375723 Square array read by antidiagonals, where the top row is the powers of 2 (A000079) and the other numbers are the sum of the neighbors in the preceding row.

Original entry on oeis.org

1, 2, 3, 4, 7, 10, 8, 14, 24, 34, 16, 28, 49, 83, 117, 32, 56, 98, 171, 288, 405, 64, 112, 196, 343, 597, 1002, 1407, 128, 224, 392, 686, 1200, 2085, 3492, 4899, 256, 448, 784, 1372, 2401, 4198, 7285, 12184, 17083, 512, 896, 1568, 2744, 4802, 8403, 14686, 25463, 42546, 59629

Offset: 0

WG Zeist, Aug 25 2024

Each number in the top row of the array is determined by the pre-defined sequence (in this case, the powers of 2, A000079). Each number in lower rows is the sum of the numbers vertically or diagonally above it (so, the number at the left end of each row is the sum of two numbers, and all other numbers the sum of three). This is the same method as for constructing A217536, which has the top row be the nonnegative integers instead; other similar arrays are described in the comments of that sequence.
The main diagonal is the powers of 7 (A000420), and all numbers above or to the right of the main diagonal are multiples of powers of 2 and powers of 7. Specifically, the number in row m and column n, for n >= m, is 2^(n-m) * 7^m. Above the main diagonal, all numbers in the same column have the same final digit in base 10, and all numbers are 7/2 times the number immediately above.
More broadly, for any similarly constructed array with the powers of x as the top row, then the main diagonal will be the powers of (x^2 + x + 1) and the numbers above the main diagonal will be x^(n-m) * (x^2 + x + 1)^m (see also A062105, which can be interpreted as a similar array with the powers of 1 in the top row, and A020474 with the powers of 0).

			The array starts:
  1  2  4   8
  3  7  14  28
  10 24 49  98
  34 83 171 343

The main diagonal gives A000420 (powers of 7). The first column gives A059738.

Cf. A217536, A062105, A020474.

T(m+1,n) = sum(T(m,k), |k-n| <= 1) (and T(0,n)=2^n), m, n >= 0.

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.

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