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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A179607 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).

Original entry on oeis.org

1, 4, 12, 56, 208, 864, 3392, 13696, 54528, 218624, 873472, 3495936, 13979648, 55926784, 223690752, 894795776, 3579117568, 14316601344, 57266143232, 229065097216, 916259340288, 3665039458304, 14660153638912, 58640622944256
Offset: 0

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Author

Johannes W. Meijer, Jul 28 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to just one red king vector, i.e., A[5] vector, with decimal [binary] value 325 [1,0,1,0,0,0,1,0,1]. This vectors leads for the corner squares to A083424 and for the side squares to A003947.
The inverse binomial transform of A100284 (without the first leading 1).

Links

Crossrefs

Cf. A179597 (central square).

Programs

  • Maple
    with(LinearAlgebra): nmax:=24; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [1,0,1,0,0,0,1,0,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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[{1},LinearRecurrence[{2,8},{4,12},30]] (* Harvey P. Dale, Mar 01 2012 *)

Formula

G.f.: (1 + 2*x - 4*x^2)/(1 - 2*x - 8*x^2).
a(n) = 2*a(n-1) + 8*a(n-2), for n >= 3, with a(0) = 1, a(1) = 4 and a(2) = 12.
a(n) = 5*(4)^(n)/6 - (-2)^(n)/3 for n >= 1 and a(0) = 1.
a(n) = 4*A083424(n-1), n>0. - R. J. Mathar, Mar 08 2021

A179609 a(n)=(5-(-1)^n-6*n)*2^(n-2).

Original entry on oeis.org

1, 0, -8, -24, -80, -192, -512, -1152, -2816, -6144, -14336, -30720, -69632, -147456, -327680, -688128, -1507328, -3145728, -6815744, -14155776, -30408704, -62914560, -134217728, -276824064, -587202560, -1207959552, -2550136832
Offset: 0

Views

Author

Johannes W. Meijer, Jul 28 2010

Keywords

Comments

This sequence belongs to a family of sequences with GF(x) = (1+(k+2)*x+(2*k-4)*x^2)/(1-2*x-(k+8)*x^2-(2*k)*x^3). Among the members of this family are several red king sequences, see A179597. For the sequence given above, which is not a red king sequence, k = -4.

Links

Programs

  • Mathematica
    Table[(5-(-1)^n-6n)2^(n-2),{n,0,30}] (* or *) LinearRecurrence[{2,4,-8},{1,0,-8},30] (* Harvey P. Dale, Mar 25 2021 *)

Formula

GF(x) = (1-2*x-12*x^2)/(1-2*x-4*x^2+8*x^3)
a(n) = 2*a(n-1)+4*a(n-2)-8*a(n-3) with a(1)=1, a(2)=0 and a(3)=-8.
a(n) = (5-(-1)^n-6*n)*2^(n-2)

A179611 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).

Original entry on oeis.org

1, 4, 16, 68, 280, 1168, 4848, 20160, 83776, 348224, 1447296, 6015488, 25002240, 103917568, 431915008, 1795179520, 7461349376, 31011794944, 128895102976, 535729963008, 2226667929600, 9254755975168, 38465775239168
Offset: 0

Views

Author

Johannes W. Meijer, Jul 28 2010

Keywords

Comments

The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596.
The sequence above corresponds to 36 red king vectors, i.e., A[5] vectors, with decimal values 15, 39, 45, 75, 78, 99, 102, 105, 108, 135, 141, 165, 195, 198, 201, 204, 225, 228, 267, 270, 291, 294, 297, 300, 330, 354, 360, 387, 390, 393, 396, 417, 420, 450, 456 and 480.

Links

Crossrefs

Cf. A179596, A179597 (central square).
Cf. A052904.

Programs

  • Maple
    with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:=[1,1,0,0,1,0,1,1,0]: A[5]:= [0,0,0,0,0,1,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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[{2,8,4},{1,4,16},30] (* Harvey P. Dale, Oct 20 2017 *)

Formula

G.f.: (1+2*x)/(1 - 2*x - 8*x^2 - 4*x^3).
a(n) = 2*a(n-1) + 8*a(n-2) + 4*a(n-3) with a(1)=1, a(2)=4 and a(3)=16.
a(n) = (8 + 3*z1 - 6*z1^2)*z1^(-n)/(z1*37) + (8 + 3*z2 - 6*z2^2)*z2^(-n)/(z2*37) + (8 + 3*z3 - 6*z3^2)*z3^(-n)/(z3*37) with z1, z2 and z3 the roots of f(x) = 1 - 2*x - 8*x^2 - 4*x^3 = 0.
alpha = arctan(3*sqrt(111));
z1 = sqrt(10)*cos(alpha/3)/6 + sqrt(30)*sin(alpha/3)/6 - 2/3 = 0.2405971520460078;
z2 = -sqrt(10)*cos(alpha/3)/3 - 2/3 = -1.585043243313016;
z3 = sqrt(10)*cos(alpha/3)/6 - sqrt(30)*sin(alpha/3)/6 - 2/3 = -0.6555539087329909.
Previous Showing 11-13 of 13 results.

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