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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A180146 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: 1/(1 - 4*x - 3*x^2 + 6*x^3).

Original entry on oeis.org

1, 4, 19, 82, 361, 1576, 6895, 30142, 131797, 576244, 2519515, 11016010, 48165121, 210591424, 920764999, 4025843542, 17602120621, 76961423116, 336496993075, 1471259517922, 6432760512217, 28125838644184, 122974079005855
Offset: 0

Views

Author

Johannes W. Meijer, Aug 13 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to 6 A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 and for the central square to A180147.

Links

Crossrefs

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

Programs

  • Maple
    with(LinearAlgebra): nmax:=22; m:=2; A[5]:=[0,1,0,1,1,1,1,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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
    Join[{a=1,b=4},Table[c=3*b+6*a+1;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)

Formula

G.f.: 1/(1 - 4*x - 3*x^2 + 6*x^3).
a(n) = 4*a(n-1) + 3*a(n-2) - 6*a(n-3) with a(-2)=0, a(-1)=0, a(0)=1, a(1)=4 and a(2)=19.
a(n) = (-1/8) + (13+30*A)*A^(-n-1)/88 + (13+30*B)*B^(-n-1)/88 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.

A180145 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 3*x^2)/(1 - 4*x - 3*x^2 + 6*x^3).

Original entry on oeis.org

1, 4, 16, 70, 304, 1330, 5812, 25414, 111112, 485818, 2124124, 9287278, 40606576, 177543394, 776269636, 3394069270, 14839825624, 64883892490, 283690631212, 1240375248574, 5423269532992, 23712060090418, 103675797469204
Offset: 0

Views

Author

Johannes W. Meijer, Aug 13 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to 6 A[5] vectors with decimal values between 191 and 506. These vectors lead for the side squares to A180146 and for the central square to A180147.

Links

Crossrefs

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

Programs

  • Maple
    with(LinearAlgebra): nmax:=22; m:=1; A[5]:=[0,1,0,1,1,1,1,1,1]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);

Formula

G.f.: (1-3*x^2)/(1 - 4*x - 3*x^2 + 6*x^3).
a(n) = 4*a(n-1) + 3*a(n-2) - 6*a(n-3) with a(0)=1, a(1)=4 and a(2)=16.
a(n) = 1/4 + (7+6*A)*A^(-n-1)/44 + (7+6*B)*B^(-n-1)/44 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.
a(n) = A180146(n) - 3*A180146(n-2) with A180146(-2) = A180146(-1) = 0.

A180144 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 - 2*x^2)/(1 - 4*x + x^2 + 2*x^3).

Original entry on oeis.org

1, 4, 13, 46, 163, 580, 2065, 7354, 26191, 93280, 332221, 1183222, 4214107, 15008764, 53454505, 190381042, 678052135, 2414918488, 8600859733, 30632416174, 109098967987, 388561736308, 1383883144897, 4928772907306
Offset: 0

Views

Author

Johannes W. Meijer, Aug 13 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6 or 8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
The sequence above corresponds to just one A[5] vector with decimal value 16. This vector leads for the corner squares to A180143 and for the central square to A000012.

Links

Crossrefs

Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).

Programs

  • Maple
    with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,1,0,0,0,0]: A:= Matrix([[0,1,1,1,0,0,1,0,0], [1,0,1,0,1,0,0,1,0], [1,1,0,0,0,1,0,0,1], [1,0,0,0,1,1,1,0,0], A[5], [0,0,1,1,1,0,0,0,1], [1,0,0,1,0,0,0,1,1], [0,1,0,0,1,0,1,0,1], [0,0,1,0,0,1,1,1,0]]): 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);

Formula

G.f.: (1-2*x^2)/(1 - 4*x + x^2 + 2*x^3).
a(n) = 4*a(n-1) - 1*a(n-2) - 2*a(n-3) with a(0)=1, a(1)=4 and a(2)=13.
a(n) = 1/4 + (21-6*A)*A^(-n-1)/68 + (21-6*B)*B^(-n-1)/68 with A=(-3+sqrt(17))/4 and B=(-3-sqrt(17))/4.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n)*(2)^(n+1)/((2*A007482(n) - 3*A007482(n-1)) - A007482(n-1)*sqrt(17)) for n >= 1.
Previous Showing 11-13 of 13 results.

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