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%I #27 Sep 08 2022 08:45:51
%S 1,3,6,14,30,66,142,306,654,1394,2958,6258,13198,27762,58254,121970,
%T 254862,531570,1106830,2301042,4776846,9903218,20505486,42409074,
%U 87614350,180821106,372827022,768023666,1580786574,3251051634
%N Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3*x^2)/(1-3*x+4*x^3).
%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 24 A[5] vectors with decimal values 7, 13, 37, 67, 70, 73, 76, 97, 100, 133, 193, 196, 259, 262, 265, 268, 289, 292, 322, 328, 352, 385, 388 and 448. These vectors lead for the side squares to A000079 and for the corner squares to A172481.
%H Vincenzo Librandi, <a href="/A175656/b175656.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 (3,0,-4).
%F G.f.: (1-3*x^2)/(1 - 3*x + 4*x^3).
%F a(n) = 3*a(n-1) - 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=6.
%F a(n) = ((3*n+22)*2^n - 4*(-1)^n)/18.
%p with(LinearAlgebra): nmax:=29; m:=5; A[5]:= [0,0,0,0,0,0,1,1,1]: 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 - 3 x^2)/(1 - 3 x + 4 x^3), {x, 0, 29}], x] (* _Michael De Vlieger_, Nov 02 2018 *)
%t LinearRecurrence[{3,0,-4},{1,3,6},30] (* _Harvey P. Dale_, Aug 12 2020 *)
%o (Magma) [((3*n+22)*2^n-4*(-1)^n)/18: n in [0..40]]; // _Vincenzo Librandi_, Aug 04 2011
%o (PARI) vector(40, n, n--; ((3*n+22)*2^n - 4*(-1)^n)/18) \\ _G. C. Greubel_, Nov 03 2018
%Y Cf. A175655 (central square).
%K nonn,easy
%O 0,2
%A _Johannes W. Meijer_, Aug 06 2010
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