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Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2^(n+2)-3*F(n+1), with F(n) = A000045(n).
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%I #20 Apr 07 2024 09:09:16

%S 1,5,10,23,49,104,217,449,922,1883,3829,7760,15685,31637,63706,128111,

%T 257353,516536,1036033,2076857,4161466,8335475,16691245,33415328,

%U 66883789,133853549,267846202,535917479,1072199137,2144987528

%N Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2^(n+2)-3*F(n+1), with F(n) = A000045(n).

%C The a(n) represent the number of n-move routes of a fairy chess piece starting in the central square (m = 5) 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 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 corner squares to A175660 (a(n)=2^(n+2)-3*F(n+2)).

%H Vincenzo Librandi, <a href="/A175661/b175661.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2).

%F G.f.: (1 + 2*x - 4*x^2)/(1 - 3*x + x^2 + 2*x^3).

%F a(n) = 3*a(n-1) - a(n-2) - 2*a(n-3) with a(0)=1, a(1)=5 and a(2)=10.

%p nmax:=29; m:=5; 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 CoefficientList[Series[(1 + 2 x - 4 x^2) / (1 - 3 x + x^2 + 2 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 21 2013 *)

%t LinearRecurrence[{3,-1,-2},{1,5,10},30] (* _Harvey P. Dale_, Apr 15 2019 *)

%o (Magma) I:=[1,5,10]; [n le 3 select I[n] else 3*Self(n-1)-Self(n-2)-2*Self(n-3): n in [1..35]]; // _Vincenzo Librandi_, Jul 21 2013

%Y Cf. A175655 (central square), A000045.

%Y Cf. A027973 (2^(n+2)+F(n)-F(n+4)), A099036 (2^n-F(n)), A167821 (2^(n+1)-2*F(n+2)), A175657 (3*2^n-2*F(n+1)), A175660 (2^(n+2)-3*F(n+2)), A179610 (convolution of (-4)^n and F(n+1)).

%K easy,nonn

%O 0,2

%A _Johannes W. Meijer_, Aug 06 2010, Aug 10 2010