A180140 -id:A180140 - cp's OEIS frontend

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

1, 2, 6, 14, 36, 86, 210, 500, 1194, 2822, 6660, 15638, 36642, 85604, 199626, 464630, 1079892, 2506550, 5811762, 13462484, 31159914, 72071654, 166599972, 384912086, 888906306, 2052031172, 4735527306, 10925175254, 25198866036, 58108609526, 133973643090

Offset: 0

Johannes W. Meijer, Aug 06 2010; edited Jun 21 2013

a(n) represents 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 bishop on the eight side and corner squares but on the center square the bishop flies into a rage and turns into a raging elephant.
In chaturanga, the old Indian version of chess, one of the pieces was called gaja, elephant in Sanskrit. The Arabs called the game shatranj and the elephant became el fil in Arabic. In Spain chess became chess as we know it today but surprisingly in Spanish a bishop isn't a Christian bishop but a Moorish elephant and it still goes by its original name of el alfil.
On a 3 X 3 chessboard there are 2^9 = 512 ways for an elephant to fly into a rage on the central square (off the center the piece behaves like a normal bishop). The elephant is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 elephants lead to 46 different elephant sequences, see the overview of elephant sequences and the crossreferences.
The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the central square to A175655.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 74, 366, 1992.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Viswanathan Anand, The Indian Defense, Time, Jun 19 2008.
Vladimir Kruchinin, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.
Johannes W. Meijer, The elephant sequences.
Wikipedia, War Elephant.
Index entries for linear recurrences with constant coefficients, signature (3,1,-6).

Cf. Elephant sequences corner squares [decimal value A[5]]: A040000 [0], A000027 [16], A000045 [1], A094373 [2], A000079 [3], A083329 [42], A027934 [11], A172481 [7], A006138 [69], A000325 [26], A045623 [19], A000129 [21], A095121 [170], A074878 [43], A059570 [15], A175654 [71, this sequence], A026597 [325], A097813 [58], A057711 [27], 2*A094723 [23; n>=-1], A002605 [85], A175660 [171], A123203 [186], A066373 [59], A015518 [341], A134401 [187], A093833 [343].

Cf. A000244, A006130, A006138, A075188, A175655, A180140.

[n le 3 select Factorial(n) else 3*Self(n-1) +Self(n-2) -6*Self(n-3): n in [1..41]]; // G. C. Greubel, Dec 08 2021

nmax:=28; m:=1; A[1]:=[0,0,0,0,1,0,0,0,1]: A[2]:=[0,0,0,1,0,1,0,0,0]: A[3]:=[0,0,0,0,1,0,1,0,0]: A[4]:=[0,1,0,0,0,0,0,1,0]: A[5]:=[0,0,1,0,0,0,1,1,1]: A[6]:=[0,1,0,0,0,0,0,1,0]: A[7]:=[0,0,1,0,1,0,0,0,0]: A[8]:=[0,0,0,1,0,1,0,0,0]: A[9]:=[1,0,0,0,1,0,0,0,0]: 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[{3,1,-6}, {1,2,6}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

a(n)=([0,1,0; 0,0,1; -6,1,3]^n*[1;2;6])[1,1] \\ Charles R Greathouse IV, Oct 03 2016

[( (1-x-x^2)/((1-2*x)*(1-x-3*x^2)) ).series(x,n+1).list()[n] for n in (0..40)] # G. C. Greubel, Dec 08 2021

G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).
a(n) = 3*a(n-1) + a(n-2) - 6*a(n-3) with a(0)=1, a(1)=2 and a(2)=6.
a(n) = ((6+10*A)*A^(-n-1) + (6+10*B)*B^(-n-1))/13 - 2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6.
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n)*2*A000244(n)/(A075118(n) - A006130(n-1)*sqrt(13)).
a(n) = b(n) - b(n-1) - b(n-2), where b(n) = Sum_{k=1..n} Sum_{j=0..k} binomial(j,n-3*k+2*j)*(-6)^(k-j)*binomial(k,j)*3^(3*k-n-j), n>0, b(0)=1, with a(0) = b(0), a(1) = b(1) - b(0). - Vladimir Kruchinin, Aug 20 2010
a(n) = 2*A006138(n) - 2^n = 2*(A006130(n) + A006130(n-1)) - 2^n. - G. C. Greubel, Dec 08 2021
E.g.f.: 2*exp(x/2)*(13*cosh(sqrt(13)*x/2) + 3*sqrt(13)*sinh(sqrt(13)*x/2))/13 - cosh(2*x) - sinh(2*x). - Stefano Spezia, Feb 12 2023

A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 11x^2 - 6*x^3).

Original entry on oeis.org

1, 3, 17, 73, 351, 1607, 7513, 34809, 161903, 751783, 3493353, 16227737, 75393055, 350251335, 1627192697, 7559508409, 35119644495, 163157037671, 757987215241, 3521419711833, 16359641017343, 76002822156295, 353090213774361

Offset: 0

Johannes W. Meijer, Jul 28 2010; edited Jun 21 2013

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 center square the king goes crazy and turns into a red king.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go crazy on the center square (off the center the piece behaves like a normal king). The red king is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 red kings lead to 47 different red king sequences, see the overview of the red king sequences.
The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [101 101 111], 463 [111 001 111], 487 [111 100 111] and 493 [111 101 101]. These vectors lead for the side squares to A126473 and for the central square to A179597.
This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 2*x - (k+8)*x^2 - 2*k*x^3). Red king sequences that are members of this family are A083424 (k=0), A179604 (k=1), A179600 (k=2), A179596 (k=3; this sequence) and A086346 (k=4). Other members of this family are A015528 (k=5) and A179608 (k=-4).

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Charles Krauthammer, Did Chess Make Him Crazy?, Time, April 26, 2005.
Johannes W. Meijer, The red king sequences.
Index entries for linear recurrences with constant coefficients, signature (2,11,6).

Cf. A180140 (berserker sequences).

Cf. Red king sequences corner squares [decimal value A[5]]: A086346 [495], A015525 [239], A179596 [367], A179600 [335], A015524 [95], A083858 [31], A179604 [327], A015523 [27], A179610 [85], A083424 [325], A015521 [11], A007482 [2], A014335 [16].

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

LinearRecurrence[{2,11,6},{1,3,17},30] (* Harvey P. Dale, May 18 2011 *)

Vec((1+x)/(1-2*x-11*x^2-6*x^3)+O(x^99)) \\ Charles R Greathouse IV, Jul 16 2011

G.f.: (1+x)/(1 - 2*x - 11*x^2 - 6*x^3).
a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0)=1, a(1)=3 and a(2)=17.
a(n) = (-1)^(-n)*2^(n+1)/9 + ((49+17*sqrt(7))*A^(-n) + (49-17*sqrt(7))*B^(-n))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*A000244(n)/(A015530(n)*sqrt(7) - A108851(n)).

A155116 a(n) = 3a(n-1) + 3a(n-2), n>2, a(0)=1, a(1)=2, a(2)=8.

Original entry on oeis.org

1, 2, 8, 30, 114, 432, 1638, 6210, 23544, 89262, 338418, 1283040, 4864374, 18442242, 69919848, 265086270, 1005018354, 3810313872, 14445996678, 54768931650, 207644784984, 787241149902, 2984657804658, 11315696863680, 42901064005014

Offset: 0

Philippe Deléham, Jan 20 2009

From Johannes W. Meijer, Aug 14 2010: (Start)
A berserker sequence, see A180140 and A180147. For the central square 16 A[5] vectors with decimal values between 3 and 384 lead to this sequence. These vectors lead for the corner squares to A123620 and for the side squares to A180142.
This sequence belongs to a family of sequences with GF(x)=(1-(2*k-1)*x-k*x^2)/(1-3*x+(k-4)*x^2). Berserker sequences that are members of this family are A000007 (k=2), A155116 (k=1; this sequence), A000302 (k=0), 6*A179606 (k=-1; with leading 1 added) and 2*A180141 (k=-2; n>=1 and a(0)=1). Some other members of this family are (-2)*A003688 (k=3; with leading 1 added), (-4)*A003946 (k=4; with leading 1 added), (-6)*A002878 (k=5; with leading 1 added) and (-8)*A033484 (k=6; with leading 1 added).
Inverse binomial transform of A101368 (without the first leading 1).
(End)

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3).

Sequences of the form a(n) = m*(a(n-1) + a(n-2)) with a(0)=1, a(1) = m-1, a(2) = m^2 -1: A155020 (m=2), this sequence (m=3), A155117 (m=4), A155119 (m=5), A155127 (m=6), A155130 (m=7), A155132 (m=8), A155144 (m=9), A155157 (m=10).

m:=3; [1] cat [n le 2 select (m-1)*(m*n-(m-1)) else m*(Self(n-1) + Self(n-2)): n in [1..30]]; // G. C. Greubel, Mar 25 2021

With[{m=3}, LinearRecurrence[{m, m}, {1, m-1, m^2-1}, 30]] (* G. C. Greubel, Mar 25 2021 *)

Vec((1-x-x^2)/(1-3*x-3*x^2)+O(x^99)) \\ Charles R Greathouse IV, Jan 12 2012

m=3; [1]+[-(m-1)*(sqrt(m)*i)^(n-2)*chebyshev_U(n, -sqrt(m)*i/2) for n in (1..30)] # G. C. Greubel, Mar 25 2021

G.f.: (1-x-x^2)/(1-3*x-3*x^2).
a(n) = 2*A125145(n-1), n>=1 .
a(n) = ( (2+4*A)*A^(-n-1) + (2+4*B)*B^(-n-1) )/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6 for n>=1 with a(0)=1. [corrected by Johannes W. Meijer, Aug 12 2010]
Contribution from Johannes W. Meijer, Aug 14 2010: (Start)
a(n) = A123620(n)/2 for n>=1.
(End)
a(n) = (1/3)*[n=0] - 2*(sqrt(3)*i)^(n-2)*ChebyshevU(n, -sqrt(3)*i/2). - G. C. Greubel, Mar 25 2021

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

Original entry on oeis.org

1, 4, 17, 71, 298, 1249, 5237, 21956, 92053, 385939, 1618082, 6783941, 28442233, 119246404, 499950377, 2096083151, 8788001338, 36844419769, 154473265997, 647641896836, 2715292020493, 11384085545659, 47728716739442

Offset: 0

Johannes W. Meijer, Jul 28 2010

a(n) represents 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 27, 30, 51, 54, 57, 60, 90, 114, 120, 147, 150, 153, 156, 177, 180, 210, 216, 240, 282, 306, 312, 402, 408 and 432. These vectors lead for the corner squares to A015523 and for the side squares to A152187.
This sequence belongs to a family of sequences with g.f. (1 + (k-4)*x)/(1 - 3*x - k*x^2). Red king sequences that are members of this family are A007483 (k= 2), A015521 (k=4), A179606 (k=5; this sequence), A154964 (k=6), A179603 (k=7) and A179599 (k=8). We observe that there is no red king sequence for k=3. Other members of this family are A006190 (k=1), A133494 (k=0) and A168616 (k=-2).
Inverse binomial transform of A052918.
The sequence b(n+1) = 6*a(n), n >= 0 with b(0)=1, is a berserker sequence, see A180147. The b(n) sequence corresponds to 16 A[5] vectors with decimal values between 111 and 492. These vectors lead for the corner squares to sequence c(n+1)=4*A179606(n), n >= 0 with c(0)=1, and for the side squares to A180140. - Johannes W. Meijer, Aug 14 2010
Equals the INVERT transform of A063782: (1, 3, 10, 32, 104, ...). Example: a(3) = 71 = (1, 1, 4, 7) dot (32, 10, 3, 1) = (32 + 10 + 12 + 17). - Gary W. Adamson, Aug 14 2010

Indranil Ghosh, Table of n, a(n) for n = 0..1603
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. A179597 (central square).

Cf. A072263, A072264, A152187, A197189.

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]:= [0,0,0,1,1,1,0,0,1]: 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+x)/(1-3*x-5*x^2), {x, 0, 22}],x] (* or *) LinearRecurrence[{3,5,0},{1,4},23] (* Indranil Ghosh, Mar 05 2017 *)

print(Vec((1 + x)/(1- 3*x - 5*x^2) + O(x^23))); \\ Indranil Ghosh, Mar 05 2017

G.f.: (1+x)/(1 - 3*x - 5*x^2).
a(n) = A015523(n) + A015523(n+1).
a(n) = 3*a(n-1) + 5*a(n-2) with a(0) = 1 and a(1) = 4.
a(n) = ((29 + 7*sqrt(29))*A^(-n-1) + (29-7*sqrt(29))*B^(-n-1))/290 with A = (-3+sqrt(29))/10 and B = (-3-sqrt(29))/10
Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A000351(n)*A130196(n)/(A015523(n)*sqrt(29) - A072263(n)) for n >= 1.

A180141 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - 2x^2)/(1 - 3x - 6*x^2).

Original entry on oeis.org

1, 4, 16, 72, 312, 1368, 5976, 26136, 114264, 499608, 2184408, 9550872, 41759064, 182582424, 798301656, 3490399512, 15261008472, 66725422488, 291742318296, 1275579489816, 5577192379224, 24385054076568, 106618316505048

Offset: 0

Johannes W. Meijer, Aug 13 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 rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go berserk on the central square (we assume here that a berserker might behave like a rook). The berserker is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the corner squares the 512 berserkers lead to 42 berserker sequences, see the cross-references for some examples.
The sequence above corresponds to just one A[5] vectors with decimal value 495. This vector leads for the side squares to 4*A154964 (for n >= 1 with a(0) = 1) and for the central square to 2*A180141 (for n >= 1 with a(0)=1).
This sequence belongs to a family of sequences with g.f. (1 + x + k*x^2)/(1 - 3*x + (k-4)*x^2), see A123620.

Index entries for linear recurrences with constant coefficients, signature (3, 6).

Cf. A180140 (side squares) and A180147 (central square).

Cf. Berserker sequences corner squares [numerical value A[5]]: 4*A055099 [0, with leading 1 added], A180143 [16], 4*A001353 [17, n>=1 and a(0)=1], A123620 [3], 2*A018916 [19, with leading 1 added], A000302 [15], 4*A179606 [111, with leading 1 added], A089979 [343], 4*A001076 [95, n>=1 and a(0)=1], A180145 [191], A180141 [495, this sequence], 4*A090017 [383, n>=1 and a(0)=1].

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

LinearRecurrence[{3, 6}, {1, 4, 16}, 23] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - 2*x^2)/(1 - 3*x - 6*x^2).
a(n) = 4*a(n-1) - 2*a(n-3) with a(0)=2, a(1)=8 and a(2)=31.
a(n) = 3*a(n-1) + 6*a(n-2) for n >= 3 with a(0)=1, a(1)=4 and a(2)=16.
a(n) = (6+2*A)*A^(-n-1)/33 + (6+2*B)*B^(-n-1)/33 with A=(-3-sqrt(33))/12 and B=(-3+sqrt(33))/12 for n >= 1 with a(0)=1.

A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 6x - 3*x^2).

Original entry on oeis.org

1, 9, 57, 369, 2385, 15417, 99657, 644193, 4164129, 26917353, 173996505, 1124731089, 7270376049, 46996449561, 303789825513, 1963728301761, 12693739287105, 82053620627913, 530402941628793, 3428578511656497

Offset: 0

Johannes W. Meijer, Aug 09 2010; edited Jun 21 2013

The a(n) represent the number of n-move routes of a fairy chess piece starting in the center square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a white 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 center square (off the center square the piece behaves like a normal queen). The red queen is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the center square the 512 red queens lead to 17 red queen sequences, see the overview of red queen sequences and the crossreferences.
The sequence above corresponds to just one red queen vector, i.e., A[5] = [111 111 111] vector. The other squares lead for this vector to A090018.
This sequence belongs to a family of sequences with g.f. (1+k*x)/(1 - 6*x - k*x^2). The members of this family that are red queen sequences are A180028 (k=3; this sequence), A180029 (k=2), A015451 (k=1), A000400 (k=0), A001653 (k=-1), A180034 (k=-2), A084120 (k=-3), A154626 (k=-4) and A000012 (k=-5). Other members of this family are A123362 (k=5), 6*A030192(k=-6).
Inverse binomial transform of A107903.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Johannes W. Meijer, The red queen sequences.
Wikipedia, Alice in Wonderland (2010 film).
Index entries for linear recurrences with constant coefficients, signature (6, 3).

Cf. A180140 (berserker sequences)
Cf. A180032 (Corner and side squares).
Cf. Red queen sequences center square [decimal value A[5]]: A180028 [511], A180029 [255], A180031 [495], A015451 [127], A152240 [239], A000400 [63], A057088 [47], A001653 [31], A122690 [15], A180034 [23], A180036 [7], A084120 [19], A180038 [3], A154626 [17], A015449 [1], A000012 [16], A000007 [0].

I:=[1,9]; [n le 2 select I[n] else 6*Self(n-1)+3*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 15 2011

nmax:=19; m:=5; A[1]:=[0,1,1,1,1,0,1,0,1]: A[2]:=[1,0,1,1,1,1,0,1,0]: A[3]:=[1,1,0,0,1,1,1,0,1]: A[4]:=[1,1,0,0,1,1,1,1,0]: A[5]:=[1,1,1,1,1,1,1,1,1]: A[6]:=[0,1,1,1,1,0,0,1,1]: A[7]:=[1,0,1,1,1,0,0,1,1]: A[8]:=[0,1,0,1,1,1,1,0,1]: A[9]:=[1,0,1,0,1,1,1,1,0]: 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[{6,3},{1,9},50] (* Vincenzo Librandi, Nov 15 2011 *)

G.f.: (1+3*x)/(1 - 6*x - 3*x^2).
a(n) = 6*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 9.
a(n) = ((1-A)*A^(-n-1) + (1-B)*B^(-n-1))/4 with A=(-1+2*sqrt(3)/3) and B=(-1-2*sqrt(3)/3).
Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n-1)*A108411(n+1)/(A041017(n-1)*sqrt(12) - A041016(n-1)) for n >= 1.

A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 4x - 3x^2 + 6x^3).

Original entry on oeis.org

1, 7, 31, 139, 607, 2659, 11623, 50827, 222223, 971635, 4248247, 18574555, 81213151, 355086787, 1552539271, 6788138539, 29679651247, 129767784979, 567381262423, 2480750497147, 10846539065983, 47424120180835

Offset: 0

Johannes W. Meijer, Aug 13 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 rook on the eight side and corner squares but on the central square the rook goes berserk and turns into a berserker, see A180140.
On a 3 X 3 chessboard there are 2^9 = 512 ways to go berserk on the central square (we assume here that a berserker might behave like a rook). The berserker is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program. For the central squares the 512 berserkers lead to 42 berserker sequences, see the cross-references for some examples.
The sequence above corresponds to six A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 and for the side squares to A180146.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4, 3, -6).

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

Cf. Berserker sequences central square [numerical values A[5]]: A000007 [0], A000012 [16], 2*A001835 [17, n>=1 and a(0)=1], A155116 [3], A077829 [7], A000302 [15], 6*A179606 [111, with leading 1 added], 2*A033887 [95, n>=1 and a(0)=1], A180147 [191, this sequence], 2*A180141 [495, n>=1 and a(0)=1], 4*A107979 [383, with leading 1 added].

with(LinearAlgebra): nmax:=22; m:=5; 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);

CoefficientList[Series[(1+3x)/(1-4x-3x^2+6x^3),{x,0,40}],x] (* or *) LinearRecurrence[{4,3,-6},{1,7,31},40] (* Harvey P. Dale, Oct 10 2011 *)

G.f.: (1+3*x)/(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)=7 and a(2)=31.
a(n) = -1/2 + (7+6*A)*A^(-n-1)/22 + (7+6*B)*B^(-n-1)/22 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.
a(n) = A180146(n) + 3*A180146(n-1) with A180146(-1) = 0.

A197189 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 11, 43, 184, 767, 3221, 13498, 56599, 237287, 994856, 4171003, 17487289, 73316882, 307387091, 1288745683, 5403172504, 22653245927, 94975600301, 398193030538, 1669457093119, 6999336432047, 29345294761736, 123032566445443, 515824173145009, 2162635351662242

Offset: 0

Bruno Berselli, Oct 11 2011

Bruno Berselli, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,5).

Cf. for type of recurrence: A015523, A072263, A072264, A152187, A179606 and also A180140.

[n le 2 select n else 3*Self(n-1)+5*Self(n-2): n in [1..26]];

a = {1, 2}; Do[AppendTo[a, 3 a[[-1]] + 5 a[[-2]]], {24}]; a (* Bruno Berselli, Dec 26 2012 *)

v=vector(26); v[1]=1; v[2]=2; for(i=3, #v, v[i]=3*v[i-1]+5*v[i-2]); v

G.f.: (1-x)/(1-3*x-5*x^2).
a(n) = ((29+sqrt(29))*(3+sqrt(29))^n+(29-sqrt(29))*(3-sqrt(29))^n)/(58*2^n).
a(n) = A015523(n+1)-A015523(n).
G.f.: G(0)*(1-x)/(2-3*x), where G(k)= 1 + 1/(1 - x*(29*k-9)/(x*(29*k+20) - 6/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 17 2013

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

Original entry on oeis.org

1, 4, 14, 54, 204, 774, 2934, 11124, 42174, 159894, 606204, 2298294, 8713494, 33035364, 125246574, 474845814, 1800277164, 6825368934, 25876938294, 98106921684, 371951579934, 1410175504854, 5346381254364, 20269670277654, 76848154596054, 291353474621124

Offset: 0

Johannes W. Meijer, Aug 13 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 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 16 A[5] vectors with decimal values between 3 and 384. These vectors lead for the corner squares to A123620 and for the central square to A155116.
This sequence appears among the members of a family of sequences with g.f. (1 + x - k*x^2)/(1 - 3*x + (k-4)*x^2). Berserker sequences that are members of this family are 4*A007482 (k=2; with leading 1 added), A180142 (k=1; this sequence), A000302 (k=0), A180140 (k=-1) and 4*A154964 (k=-2; n>=1 and a(0)=1). Some other members of this family are 2*A180148 (k=3; with leading 1 added), 4*A025192 (k=4; with leading 1 added), 2*A005248 (k=5; with leading 1 added) and A123932 (k=6).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,3).

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

with(LinearAlgebra): nmax:=23; m:=2; A[5]:=[0,0,0,0,0,0,0,1,1]: 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);
# second Maple program:
a:= n-> ceil((<<0|1>, <3|3>>^n. <<2/3, 4>>)[1,1]):
seq(a(n), n=0..25);  # Alois P. Heinz, Jul 14 2021

LinearRecurrence[{3, 3}, {1, 4, 14}, 26] (* Jean-François Alcover, Jan 18 2025 *)

G.f.: (1 + x - x^2)/(1 - 3*x - 3*x^2).
a(n) = 3*a(n-1) + 3*a(n-2) for n >= 2 with a(0)=1, a(1)=4 and a(2)=14.
a(n) = (6-2*A)*A^(-n-1)/21 + (6-2*B)*B^(-n-1)/21 with A=(-3+sqrt(21))/6 and B=(-3-sqrt(21))/6.
Lim_{k->infinity} a(2*n+k)/a(k) = 2*A000244(n)/(A003501(n) - A004254(n)*sqrt(21)) for n >= 1.
Lim_{k->infinity} a(2*n-1+k)/a(k) = 2*A000244(n)/(A004253(n)*sqrt(21) - 3*A030221(n-1)) for n >= 1.

A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4x + x^2 + 2x^3).

Original entry on oeis.org

1, 4, 16, 58, 208, 742, 2644, 9418, 33544, 119470, 425500, 1515442, 5397328, 19222870, 68463268, 243835546, 868433176, 3092970622, 11015778220, 39233275906, 139731384160, 497660704294, 1772444881204, 6312656052202

Offset: 0

Johannes W. Meijer, Aug 13 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 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] vectors with decimal value 16. This vector leads for the side squares to A180144 and for the central square to A000012.

Index entries for linear recurrences with constant coefficients, signature (4, -1, -2).

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

with(LinearAlgebra): nmax:=23; m:=1; 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);

G.f.: (1+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)=16.
a(n) = -1/2 + (9+12*A)*A^(-n-1)/34 + (9+12*B)*B^(-n-1)/34 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.

cp's OEIS Frontend

A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3*x - x^2 + 6*x^3).

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A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 11*x^2 - 6*x^3).

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A155116 a(n) = 3*a(n-1) + 3*a(n-2), n>2, a(0)=1, a(1)=2, a(2)=8.

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A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3*x - 5*x^2).

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A180141 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - 2*x^2)/(1 - 3*x - 6*x^2).

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A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 6*x - 3*x^2).

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A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3*x)/(1 - 4*x - 3*x^2 + 6*x^3).

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A175654 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1 - x - x^2)/(1 - 3x - x^2 + 6x^3).

A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2x - 11x^2 - 6*x^3).

A155116 a(n) = 3a(n-1) + 3a(n-2), n>2, a(0)=1, a(1)=2, a(2)=8.

A179606 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 3x - 5x^2).

A180141 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - 2x^2)/(1 - 3x - 6*x^2).

A180028 Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 6x - 3*x^2).

A180147 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + 3x)/(1 - 4x - 3x^2 + 6x^3).

A197189 a(n) = 3a(n-1) + 5a(n-2), with a(0)=1, a(1)=2.

A180142 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x - x^2)/(1 - 3x - 3x^2).

A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4x + x^2 + 2x^3).