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

%I A180143 #6 Jul 03 2023 11:39:43
%S A180143 1,4,16,58,208,742,2644,9418,33544,119470,425500,1515442,5397328,
%T A180143 19222870,68463268,243835546,868433176,3092970622,11015778220,
%U A180143 39233275906,139731384160,497660704294,1772444881204,6312656052202
%N A180143 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: (1 + x^2)/(1 - 4*x + x^2 + 2*x^3).
%C A180143 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 A180143 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.
%H A180143 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, -1, -2).
%F A180143 G.f.: (1+x^2)/(1 - 4*x + x^2 + 2*x^3).
%F A180143 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.
%F A180143 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.
%F A180143 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.
%p A180143 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);
%Y A180143 Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).
%K A180143 easy,nonn
%O A180143 0,2
%A A180143 _Johannes W. Meijer_, Aug 13 2010