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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 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

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