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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).

%I A180145 #6 Jul 03 2023 12:30:54
%S A180145 1,4,16,70,304,1330,5812,25414,111112,485818,2124124,9287278,40606576,
%T A180145 177543394,776269636,3394069270,14839825624,64883892490,283690631212,
%U A180145 1240375248574,5423269532992,23712060090418,103675797469204
%N 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).
%C A180145 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 A180145 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.
%H A180145 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, 3, -6).
%F A180145 G.f.: (1-3*x^2)/(1 - 4*x - 3*x^2 + 6*x^3).
%F A180145 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.
%F A180145 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.
%F A180145 a(n) = A180146(n) - 3*A180146(n-2) with A180146(-2) = A180146(-1) = 0.
%p A180145 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);
%Y A180145 Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).
%K A180145 easy,nonn
%O A180145 0,2
%A A180145 _Johannes W. Meijer_, Aug 13 2010