A180146 -id:A180146 - cp's OEIS frontend

A180140 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1+x+x^2)/(1-3x-5x^2).

1, 4, 18, 74, 312, 1306, 5478, 22964, 96282, 403666, 1692408, 7095554, 29748702, 124723876, 522915138, 2192364794, 9191670072, 38536834186, 161568852918, 677390729684, 2840016453642, 11907003009346, 49921091296248

Offset: 0

Johannes W. Meijer, Aug 13 2010, Jun 15 2013

a(n) gives the number of n-move routes of a fairy chess piece starting in a given side square (m = 2, 4, 6,  8) on a 3 X 3 chessboard. This fairy chess piece behaves like a rook on the four side and four corner (m = 1, 3, 7, 9) squares but on the center square (m = 5) it goes berserk and turns into a berserker. For this sequence, the berserker can move to three of the side squares and three of the corners from the center.
The berserker is one of the Lewis chessmen which were discovered in 1831 on the Isle of Lewis. They are carved from walrus ivory in Scandinavian style of the 12th century. The pawns look like decorated tombstones. The pieces have all human representations with facial expressions varying from gloom to anger. Some of the rooks show men biting their shield in the manner of berserkers. According to Hooper and Whyld none looks happy.
Let A be the adjacency matrix of the graph G, where V(G) = {v1, v2, v3, v4, v5, v6, v7, v8, v9}. Then the (m, k) entry of A^n is the number of different vm-vk walks of length n in G, see the Chartrand reference. In the adjacency matrix A, see the Maple program, the A[1], A[3], A[7] and A[9] vectors represent the rook moves on the corner squares, the A[2], A[4], A[6] and A[8] vectors represent the rook moves on the side squares and the A[5] vector represents the moves of the berserker. On a 3 X 3 chessboard there are 2^9 = 512 ways a berserker could move from the center square (off the center the berserker behaves like a rook) so there are 512 different berserkers.
For the side squares the 512 berserker vectors lead to 42 different sequences, see the overview of berserker sequences. There are 16 berserker vectors that lead to the sequence given above. Their decimal [binary] values are: 111 [001 101 111] , 207 [011 001 111], 231 [011 100 111], 237 [011 101 101], 303 [100 101 111], 363 [101 101 011], 366 [101 101 110], 399 [110 001 111], 423 [110 100 111], 429 [110 101 101], 459 [111 001 011], 462 [111 001 110], 483 [111 100 011], 486 [111 100 110], 489 [111 101 001] and 492 [111 101 100]. These berserker vectors lead for the corner squares to sequence 4*A179606 (with leading term 1 added) and for the central square to sequence 6*A179606 (with leading term 1 added).
This sequence belongs to a family of sequences with GF(x)=(1+x-k*x^2)/(1-3*x+(k-4)*x^2), see A180142.

Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
David Hooper and Kenneth Whyld, The Oxford Companion to Chess, pp. 131, 225, 1992.

Indranil Ghosh, Table of n, a(n) for n = 0..1603
Dougie MacLean, The Lewis Chessmen: "Marching Mystery"
Johannes W. Meijer, The berserker sequences.
Wikipedia, Lewis Chessmen
Index entries for linear recurrences with constant coefficients, signature (3,5)

Cf. A180141 (corner squares) and A180147 (central square).

Cf. Berserker sequences side squares: 4*A007482 (with leading 1 added), A180144, A003500 (n>=1 and a(0)=1), A180142, A000302, A180140 (this sequence), 2*A001077 (n>=1 and a(0)=1), A180146, 4*A154964 (n>=1 and a(0)=1), 4*A123347 (with leading 1 added).

nmax:=22; m:=2; A[1]:=[0, 1, 1, 1, 0, 0, 1, 0, 0]: A[2]:=[1, 0, 1, 0, 1, 0, 0, 1, 0]: A[3]:= [1, 1, 0, 0, 0, 1, 0, 0, 1]: A[4]:= [1, 0, 0, 0, 1, 1, 1, 0, 0]: A[5]:=[0, 0, 1, 1, 0, 1, 1, 1, 1]: A[6]:=[0, 0, 1, 1, 1, 0, 0, 0, 1]: A[7]:=[1, 0, 0, 1, 0, 0, 0, 1, 1]: A[8]:=[0, 1, 0, 0, 1, 0, 1, 0, 1]: A[9]:=[0, 0, 1, 0, 0, 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);

CoefficientList[Series[(1+x+x^2)/(1-3*x-5*x^2), {x, 0, 22}],x] (* or *) LinearRecurrence[{3,5,0},{1,4,18},23] (* Indranil Ghosh, Mar 05 2017 *)

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

G.f.: (1+x+x^2)/(1-3*x-5*x^2).
a(n) = 3*a(n-1) + 5*a(n-2) for n>=3  with a(0)=1, a(1)=4 and a(2)=18.
a(n) = ((22+54*A)*A^(-n-1) + (22+54*B)*B^(-n-1))/145  with A=(-3+sqrt(29))/10 and B=(-3-sqrt(29))/10 for n>=1 with a(0)=1.
5*a(n) = 2*( A015523(n) + 3*A015523(n+1)), n>0 - R. J. Mathar, May 11 2013

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.

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

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 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.

Index entries for linear recurrences with constant coefficients, signature (4, 3, -6).

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

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);

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.

A110291 Riordan array (1/(1-x), x(1+2x)).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 3, 9, 7, 1, 1, 3, 9, 19, 9, 1, 1, 3, 9, 27, 33, 11, 1, 1, 3, 9, 27, 65, 51, 13, 1, 1, 3, 9, 27, 81, 131, 73, 15, 1, 1, 3, 9, 27, 81, 211, 233, 99, 17, 1, 1, 3, 9, 27, 81, 243, 473, 379, 129, 19, 1, 1, 3, 9, 27, 81, 243, 665, 939, 577, 163, 21, 1

Offset: 0

Paul Barry, Jul 18 2005

Inverse is A110292.

			Rows begin
  1;
  1, 1;
  1, 3, 1;
  1, 3, 5,  1;
  1, 3, 9,  7,  1;
  1, 3, 9, 19,  9,   1;
  1, 3, 9, 27, 33,  11,  1;
  1, 3, 9, 27, 65,  51, 13,  1;
  1, 3, 9, 27, 81, 131, 73, 15, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000244, A001045, A001047, A005408, A014335.
Cf. A066810, A077890, A077912, A110292, A180146.
Cf. A000975 (row sums), A052947 (diagonal sums).

R:=PowerSeriesRing(Rationals(), 30);
F:= func< k | Coefficients(R!( x^k*(1+2*x)^k/(1-x) )) >;
A110291:= func< n,k | F(k)[n-k+1] >;
[A110291(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 05 2023

F[k_]:= CoefficientList[Series[x^k*(1+2*x)^k/(1-x), {x,0,40}], x];
A110291[n_, k_]:= F[k][[n+1]];
Table[A110291[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 05 2023 *)

def p(k,x): return x^k*(1+2*x)^k/(1-x)
def A110291(n,k): return ( p(k,x) ).series(x, 30).list()[n]
flatten([[A110291(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 05 2023

T(n, k) = [x^n]( x^k*(1+2*x)^k/(1-x) ).
Sum_{k=0..n} T(n, k) = A000975(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A052947(n+1).
From G. C. Greubel, Jan 05 2023: (Start)
T(n, 0) = T(n, n) = 1.
T(n, n-1) = A005408(n-1).
T(2*n, n) = T(2*n+1, n) = A000244(n).
T(2*n, n+1) = A066810(n+1).
T(2*n, n-1) = A000244(n-1).
T(2*n+1, n+1) = A001047(n+1).
Sum_{k=0..n} (-1)^k * T(n, k) = A077912(n).
Sum_{k=0..n} 2^k * T(n, k) = A014335(n+2).
Sum_{k=0..n} 3^k * T(n, k) = A180146(n).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A077890(n). (End)

a(30) and following corrected by Georg Fischer, Oct 11 2022

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A180140 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1+x+x^2)/(1-3*x-5*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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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).

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A110291 Riordan array (1/(1-x), x*(1+2*x)).

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A180140 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1+x+x^2)/(1-3x-5x^2).

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

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

A110291 Riordan array (1/(1-x), x(1+2x)).