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A180146 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: 1/(1 - 4*x - 3*x^2 + 6*x^3).

%I A180146 #11 Jul 03 2023 12:31:44
%S A180146 1,4,19,82,361,1576,6895,30142,131797,576244,2519515,11016010,
%T A180146 48165121,210591424,920764999,4025843542,17602120621,76961423116,
%U A180146 336496993075,1471259517922,6432760512217,28125838644184,122974079005855
%N A180146 Eight rooks and one berserker on a 3 X 3 chessboard. G.f.: 1/(1 - 4*x - 3*x^2 + 6*x^3).
%C A180146 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.
%C A180146 The sequence above corresponds to 6 A[5] vectors with decimal values between 191 and 506. These vectors lead for the corner squares to A180145 and for the central square to A180147.
%H A180146 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4, 3, -6).
%F A180146 G.f.: 1/(1 - 4*x - 3*x^2 + 6*x^3).
%F A180146 a(n) = 4*a(n-1) + 3*a(n-2) - 6*a(n-3) with a(-2)=0, a(-1)=0, a(0)=1, a(1)=4 and a(2)=19.
%F A180146 a(n) = (-1/8) + (13+30*A)*A^(-n-1)/88 + (13+30*B)*B^(-n-1)/88 with A=(-3+sqrt(33))/12 and B=(-3-sqrt(33))/12.
%p A180146 with(LinearAlgebra): nmax:=22; m:=2; 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);
%t A180146 Join[{a=1,b=4},Table[c=3*b+6*a+1;a=b;b=c,{n,60}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 18 2011 *)
%Y A180146 Cf. A180141 (corner squares), A180140 (side squares), A180147 (central square).
%K A180146 easy,nonn
%O A180146 0,2
%A A180146 _Johannes W. Meijer_, Aug 13 2010