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A175660 Eight bishops and one elephant on a 3 X 3 chessboard. a(n) = 2^(n+2) - 3*F(n+2).

%I A175660 #10 Nov 23 2018 16:25:02
%S A175660 1,2,7,17,40,89,193,410,859,1781,3664,7493,15253,30938,62575,126281,
%T A175660 254392,511745,1028281,2064314,4141171,8302637,16638112,33329357,
%U A175660 66744685,133628474,267482023,535328225,1071245704,2143444841
%N A175660 Eight bishops and one elephant on a 3 X 3 chessboard. a(n) = 2^(n+2) - 3*F(n+2).
%C A175660 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, 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654.
%C A175660 The sequence above corresponds to four A[5] vectors with decimal values 171, 174, 234 and 426. These vectors lead for the side squares to A000079 and for the central square to A175661 (a(n) = 2^(n+2) - 3*F(n+1)).
%H A175660 Harvey P. Dale, <a href="/A175660/b175660.txt">Table of n, a(n) for n = 0..1000</a>
%H A175660 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3, -1, -2).
%F A175660 G.f.: (1 - x + 2*x^2)/(1 - 3*x + x^2 + 2*x^3).
%F A175660 a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) with a(0)=1, a(1)=2 and a(2)=7.
%F A175660 a(n) = 2^(n+2) - 3*F(n+2) with F(n)=A000045(n).
%p A175660 nmax:=29; m:=1; A[5]:= [0,1,0,1,0,1,0,1,1]: A:=Matrix([[0,0,0,0,1,0,0,0,1], [0,0,0,1,0,1,0,0,0], [0,0,0,0,1,0,1,0,0], [0,1,0,0,0,0,0,1,0], A[5], [0,1,0,0,0,0,0,1,0], [0,0,1,0,1,0,0,0,0], [0,0,0,1,0,1,0,0,0], [1,0,0,0,1,0,0,0,0]]): 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 A175660 Table[2^(n+2)-3Fibonacci[n+2],{n,0,30}] (* or *) LinearRecurrence[ {3,-1,-2},{1,2,7},30] (* _Harvey P. Dale_, Dec 28 2012 *)
%Y A175660 Cf. A008466 (2^n-F(n+2)), A027934 (2^n-F(n+1)), A027974 (2^(n+3)-F(n+5)-F(n+3)), A074878 (3*2^n-2*F(n+2)), A142585 ((-1)^(n+1)*(2^(n-1)-F(n+1)-F(n-1))), A175661 (2^(n+2)-3*F(n+1)), A179610 (convolution of (-4)^n and F(n+1)).
%K A175660 easy,nonn
%O A175660 0,2
%A A175660 _Johannes W. Meijer_, Aug 06 2010, Aug 10 2010