cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A180032 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1+x)/(1-5*x-7*x^2).

Original entry on oeis.org

1, 6, 37, 227, 1394, 8559, 52553, 322678, 1981261, 12165051, 74694082, 458625767, 2815987409, 17290317414, 106163498933, 651849716563, 4002393075346, 24574913392671, 150891318490777, 926480986202582, 5688644160448349
Offset: 0

Views

Author

Johannes W. Meijer, Aug 09 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner or side square (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a white chess queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen.
On a 3 X 3 chessboard there are 2^9 = 512 ways to explode with fury on the central square (we assume here that a red queen might behave like a white queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner and side squares the 512 red queens lead to 17 red queen sequences, see the cross-references for the complete set.
The sequence above corresponds to 8 red queen vectors, i.e., A[5] vectors, with decimal values 239, 367, 431, 463, 487, 491, 493 and 494. The central square leads for these vectors to A152240.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 5*x - k*x^2). The members of this family that are red queen sequences are A180030 (k=8), A180032 (k=7; this sequence), A000400 (k=6), A180033 (k=5), A126501 (k=4), A180035 (k=3), A180037 (k=2) A015449 (k=1) and A003948 (k=0). Other members of this family are A030221 (k=-1), A109114 (k=-3), A020989 (k=-4), A166060 (k=-6).
Inverse binomial transform of A054413.

Links

Crossrefs

Cf. A180028 (Central square).
Cf. Red queen sequences corner and side squares [decimal value A[5]]: A090018 [511], A135030 [255], A180030 [495], A005668 [127], A180032 [239], A000400 [63], A180033 [47], A001109 [31], A126501 [15], A154244 [23], A180035 [7], A138395 [19], A180037 [3], A084326 [17], A015449 [1], A003463 [16], A003948 [0].

Programs

  • Magma
    I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+7*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011
  • Maple
    with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [1,1,1,1,0,1,1,1,0]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);
  • Mathematica
    LinearRecurrence[{5,7},{1,6},40] (* Vincenzo Librandi, Nov 15 2011 *)
CoefficientList[Series[(1+x)/(1-5x-7x^2),{x,0,30}],x] (* Harvey P. Dale, Apr 04 2024 *)

Formula

G.f.: (1+x)/(1 - 5*x - 7*x^2).
a(n) = 5*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((7+9*A)*A^(-n-1) + (7+9*B)*B^(-n-1))/53 with A = (-5+sqrt(53))/14 and B = (-5-sqrt(53))/14.

A180031 Number of n-move paths on a 3 X 3 chessboard of a queen starting or ending in the central square.

Original entry on oeis.org

1, 8, 48, 304, 1904, 11952, 74992, 470576, 2952816, 18528688, 116265968, 729559344, 4577924464, 28726097072, 180253881072, 1131078181936, 7097421958256, 44535735246768, 279458051899888, 1753576141473584
Offset: 0

Views

Author

Johannes W. Meijer, Aug 09 2010

Keywords

Comments

The a(n) represent the number of n-move paths of a chess queen starting or ending in the central square (m = 5) on a 3 X 3 chessboard. The other squares lead to A180030.
To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the queen's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
Closely related with this sequence are the red queen sequences, see A180028 and A180032.
This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 5*x - (k+5)*x^2). The members of this family that are red queen sequences are A180031 (k=3; this sequence), A152240 (k=2), A000400 (k=1), A057088 (k=0), A122690 (k=-1), A180036 (k=-2), A180038 (k=-3), A015449 (k=-4) and A000007 (k=-5). Other members of this family are A030221 (k= -6), 3*A109114 (k=-8), 4*A020989 (k=-9), 6*A166060 (k=-11).

Links

Programs

  • Magma
    I:=[1,8]; [n le 2 select I[n] else 5*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011
  • Maple
    with(LinearAlgebra): nmax:=19; m:=5; A[5]:= [1,1,1,1,0,1,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): for n from 0 to nmax do B(n):=A^n: a(n):= add(B(n)[m,k],k=1..9): od: seq(a(n), n=0..nmax);
  • Mathematica
    LinearRecurrence[{5,8},{1,8},50] (* Vincenzo Librandi, Nov 15 2011 *)

Formula

G.f.: (1+3*x)/(1 - 5*x - 8*x^2).
a(n) = 5*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 8.
a(n) = ((A+11)*A^(-n-1) + (B+11)*B^(-n-1))/57 with A = (-5+sqrt(57))/16 and B = (-5-sqrt(57))/16.
Showing 1-2 of 2 results.

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