A179596 -id:A179596 - cp's OEIS frontend

A179599 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 4x)/(1 - 3x - 8*x^2).

1, 7, 29, 143, 661, 3127, 14669, 69023, 324421, 1525447, 7171709, 33718703, 158529781, 745338967, 3504255149, 16475477183, 77460472741, 364185235687, 1712239488989, 8050200352463, 37848516969301, 177947153727607

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 10 red king vectors, i.e., A[5] vectors, with decimal values 239, 351, 375, 381, 431, 471, 477, 491, 494 and 501. These vectors lead for the corner squares to A015525 and for the side squares to A179598.

Index entries for linear recurrences with constant coefficients, signature (3, 8).

Cf. A179597 (central square).

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]:= [1,0,1,1,1,1,1,0,1]: 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);

G.f.: (1+4*x)/(1 - 3*x - 8*x^2).
a(n) = 3*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 7.
a(n) = ((11+4*A)*A^(-n-1) + (11+4*B)*B^(-n-1))/41 with A = (-3+sqrt(41))/16 and B = (-3-sqrt(41))/16.

A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x)/(1 - 3x - 7*x^2).

Original entry on oeis.org

1, 5, 22, 101, 457, 2078, 9433, 42845, 194566, 883613, 4012801, 18223694, 82760689, 375847925, 1706868598, 7751541269, 35202703993, 159868900862, 726025630537, 3297159197645, 14973657006694, 68001085403597, 308818855257649

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 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 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the central square to A179603.

Index entries for linear recurrences with constant coefficients, signature (3, 7).

Cf. A126473 (side squares).

with(LinearAlgebra): nmax:=22; m:=2; 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]:= [1,1,1,0,1,0,0,1,1]: 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);

CoefficientList[Series[(1+2x)/(1-3x-7x^2),{x,0,40}],x] (* or *) LinearRecurrence[ {3,7},{1,5},40] (* Harvey P. Dale, Mar 28 2013 *)

G.f.: (1+2*x)/(1 - 3*x - 7*x^2).
a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 5.
a(n) = ((37+4*37^(1/2))*A^(-n-1) + (37-4*37^(1/2))*B^(-n-1))/259 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.

A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 3x - 7*x^2).

Original entry on oeis.org

1, 6, 25, 117, 526, 2397, 10873, 49398, 224305, 1018701, 4626238, 21009621, 95412529, 433304934, 1967802505, 8936542053, 40584243694, 184308525453, 837015282217, 3801205524822, 17262723549985, 78396609323709

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the side squares to A179602.

Index entries for linear recurrences with constant coefficients, signature (3, 7).

Cf. A179597 (central square).

with(LinearAlgebra): nmax:=23; 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]:= [1,1,1,0,1,0,0,1,1]: 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);

G.f.: (1+3*x)/(1 - 3*x - 7*x^2).
a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 6.
a(n) = ((9+5*A)*A^(-n-1) + (9+5*B)*B^(-n-1))/37 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14.

A179608 a(n) = (7 + (-1)^n + 6n)2^(n-3).

Original entry on oeis.org

1, 3, 10, 24, 64, 144, 352, 768, 1792, 3840, 8704, 18432, 40960, 86016, 188416, 393216, 851968, 1769472, 3801088, 7864320, 16777216, 34603008, 73400320, 150994944, 318767104, 654311424, 1375731712, 2818572288, 5905580032

Offset: 0

Johannes W. Meijer, Jul 28 2010

This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 2*x - (k+8)*x^2 - 2*k*x^3). Among the members of this family are several red king sequences, see A179596. For the sequence given above, which is not a red king sequence, k = -4.

Inverse binomial transform of A119916 (without the leading 0).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2, 4, -8).

[(7+(-1)^n+6*n)*2^(n-3): n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

G.f.: (1+x)/(1 - 4*x^2 - 2*x + 8*x^3).
a(n) = 2*a(n-1) + 4*a(n-2) - 8*a(n-3) with a(1)=1, a(2)=3 and a(3)=10.
a(n) = (7 + (-1)^n + 6*n)*2^(n-3).

A179610 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: 1/(1-3x-5x^2+4*x^3).

Original entry on oeis.org

1, 3, 14, 53, 217, 860, 3453, 13791, 55198, 220737, 883037, 3532004, 14128249, 56512619, 226051086, 904203357, 3616815025, 14467257516, 57869034245, 231476130215, 925904531806, 3703618109513, 14814472466709, 59257889820468

Offset: 0

Johannes W. Meijer, Jul 28 2010, revised Aug 15 2010

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 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 4 red king vectors, i.e. A[5] vectors, with decimal [binary] values 85 [0,0,1,0,1,0,1,0,1], 277 [1,0,0,0,1,0,1,0,1], 337 [1,0,1,0,1,0,0,0,1] and 340 [1,0,1,0,1,0,1,0,0].
Convolution of (-4)^n and F(n+1) with F = A000045.

Index entries for linear recurrences with constant coefficients, signature (3, 5, -4).

with(LinearAlgebra): nmax:=23; m:=1; 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]:= [1,0,0,0,1,0,1,0,1]: 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);

CoefficientList[Series[1/(1-3x-5x^2+4x^3),{x,0,30}],x] (* or *) LinearRecurrence[{3,5,-4},{1,3,14},30] (* Harvey P. Dale, Aug 12 2025 *)

G.f.: = 1/((x^2-x-1)*(4*x-1)).
a(n) = 3*a(n-1)+5*a(n-2)-4*a(n-3) with a(1)=1, a(2)=3 and a(3)=14.
a(n) = (1/95)*(5*2^(2*n+4)-(11-2*phi)*phi^(-n-1)-(9+2*phi)*(1-phi)^(-n-1)) with phi = (1+sqrt(5))/2, with A001622 = phi.
a(n) = (-1)^n*sum((-4)^m*F(n+1-m),m=0..n).

A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 10x^2 - 4*x^3).

Original entry on oeis.org

1, 3, 16, 66, 304, 1332, 5968, 26472, 117952, 524496, 2334400, 10385568, 46213120, 205619520, 914912512, 4070872704, 18113348608, 80595074304, 358607125504, 1595618388480, 7099688329216, 31589989045248, 140559334936576

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 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 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the side squares to A123347 and for the central square to A179601.

Index entries for linear recurrences with constant coefficients, signature (2,10,4).

with(LinearAlgebra): nmax:=24; m:=1; 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]:= [1,1,1,0,0,0,1,1,1]: 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);

Vec((1+x)/(1 - 2*x - 10*x^2 - 4*x^3) + O(x^40)) \\ Jinyuan Wang, Mar 10 2020

G.f.: (1+x)/(1 - 2*x - 10*x^2 - 4*x^3).
a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=16.
a(n) = (4*(-1/2)^(-n) + (1+sqrt(6))*A^(-n-1) + (1-sqrt(6))*B^(-n-1))/20 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*(A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1))) for n => 1.

A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4x)/(1 - 2x - 10x^2 - 4x^3).

Original entry on oeis.org

1, 6, 22, 108, 460, 2088, 9208, 41136, 182704, 813600, 3618784, 16104384, 71651008, 318820992, 1418569600, 6311953152, 28084886272, 124963582464, 556023840256, 2474023050240, 11008138832896, 48980603529216, 217938687588352

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the corner squares to A179600 and for the side squares to A123347.

Index entries for linear recurrences with constant coefficients, signature (2,10,4).

Cf. A041006, A041007, A123347, A179596, A179597 (central square), A179600.

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]:= [1,1,1,0,0,0,1,1,1]: 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);

G.f.: ( -1-4*x ) / ( (2*x+1)*(2*x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=6 and a(2)=22.
a(n) = (-2/5)*(-1/2)^(-n) + ((2+3*A)*A^(-n-1) + (2+3*B)*B^(-n-1))/10 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1)) for n => 1.

A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 9x^2 - 2*x^3).

Original entry on oeis.org

1, 3, 15, 59, 259, 1079, 4607, 19443, 82507, 349215, 1479879, 6267707, 26552755, 112474631, 476459471, 2018296131, 8549676763, 36216937647, 153417558423, 649886909195, 2752965719491, 11661748738583, 49399962770975

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 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 4 red king vectors, i.e., A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the side squares to A015448 and for the central square to A179605.

Index entries for linear recurrences with constant coefficients, signature (2,9,2).

Cf. A001076, A001077, A015448, A179605, A179596.

with(LinearAlgebra): nmax:=22; m:=1; 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]:= [1,0,1,1,0,0,1,0,1]: 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);

LinearRecurrence[{2,9,2},{1,3,15},30] (* or *) CoefficientList[ Series[ (x+1)/(-2 x^3-9 x^2-2 x+1),{x,0,30}],x] (* Harvey P. Dale, Mar 17 2012 *)

G.f.: ( -1-x ) / ( (2*x+1)*(x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=3 and a(2)=15.
a(n) = (20*(-1/2)^(-n) + (5+7*sqrt(5))*A^(-n-1) + (5-7*sqrt(5))*B^(-n-1))/110 with A = (-2+sqrt(5)) and B:= (-2-sqrt(5)).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).

A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x - 2x^2)/(1 - 2x - 9x^2 - 2*x^3).

Original entry on oeis.org

1, 5, 17, 81, 325, 1413, 5913, 25193, 106429, 451421, 1911089, 8097825, 34298293, 145299189, 615478665, 2607246617, 11044399597, 46784976077, 198184041761, 839521667409, 3556269662821, 15064602415845, 63814675131897

Offset: 0

Johannes W. Meijer, Jul 28 2010

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 toes crazy and turns into a red king, see A179596.

The sequence above corresponds to 4 red king vectors, A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the corner squares to A179604 and for the side squares to A015448.

Index entries for linear recurrences with constant coefficients, signature (2,9,2).

Cf. A001076, A001077, A015448, A179596, A179597 (central square), A179604.

with(LinearAlgebra): nmax:=21; 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]:= [1,0,1,1,0,0,1,0,1]: 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);

G.f.: ( -1 - 3*x + 2*x^2 ) / ( (2*x+1)*(x^2 + 4*x - 1) ).
a(n) = 2*a(n-1) + 9*a(n-2) + 2*a(n-3) with a(0)=1, a(1)=5 and a(2)=17.
a(n) = (-4/11)*(-1/2)^(-n) + ((17+41*A)*A^(-n-1) + (17+41*B)*B^(-n-1))/110 with A = (-2+sqrt(5)) and B =(-2-sqrt(5)).
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)/(A001076(n)*sqrt(5) - A001077(n)).

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

Johannes W. Meijer, Jul 28 2010

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).

Index entries for linear recurrences with constant coefficients, signature (2, 8).

Cf. A179597 (central square).

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);

Join[{1},LinearRecurrence[{2,8},{4,12},30]] (* Harvey P. Dale, Mar 01 2012 *)

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

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A179599 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 4*x)/(1 - 3*x - 8*x^2).

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A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 3*x - 7*x^2).

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A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 3*x - 7*x^2).

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A179608 a(n) = (7 + (-1)^n + 6*n)*2^(n-3).

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A179610 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: 1/(1-3*x-5*x^2+4*x^3).

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A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 10*x^2 - 4*x^3).

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A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4*x)/(1 - 2*x - 10*x^2 - 4*x^3).

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A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 9*x^2 - 2*x^3).

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A179599 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 4x)/(1 - 3x - 8*x^2).

A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2x)/(1 - 3x - 7*x^2).

A179603 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 3x - 7*x^2).

A179608 a(n) = (7 + (-1)^n + 6n)2^(n-3).

A179610 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: 1/(1-3x-5x^2+4*x^3).

A179600 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 10x^2 - 4*x^3).

A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4x)/(1 - 2x - 10x^2 - 4x^3).

A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 9x^2 - 2*x^3).

A179605 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 3x - 2x^2)/(1 - 2x - 9x^2 - 2*x^3).

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