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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).

%I A180141 #10 Jan 18 2025 09:08:26
%S A180141 1,4,16,72,312,1368,5976,26136,114264,499608,2184408,9550872,41759064,
%T A180141 182582424,798301656,3490399512,15261008472,66725422488,291742318296,
%U A180141 1275579489816,5577192379224,24385054076568,106618316505048
%N 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).
%C A180141 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.
%C A180141 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.
%C A180141 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).
%C A180141 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.
%H A180141 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (3, 6).
%F A180141 G.f.: (1 + x - 2*x^2)/(1 - 3*x - 6*x^2).
%F A180141 a(n) = 4*a(n-1) - 2*a(n-3) with a(0)=2, a(1)=8 and a(2)=31.
%F A180141 a(n) = 3*a(n-1) + 6*a(n-2) for n >= 3 with a(0)=1, a(1)=4 and a(2)=16.
%F A180141 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.
%p A180141 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);
%t A180141 LinearRecurrence[{3, 6}, {1, 4, 16}, 23] (* _Jean-François Alcover_, Jan 18 2025 *)
%Y A180141 Cf. A180140 (side squares) and A180147 (central square).
%Y A180141 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].
%K A180141 easy,nonn
%O A180141 0,2
%A A180141 _Johannes W. Meijer_, Aug 13 2010