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A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3*x^2)/(1-3*x+4*x^3).

%I A175656 #27 Sep 08 2022 08:45:51
%S A175656 1,3,6,14,30,66,142,306,654,1394,2958,6258,13198,27762,58254,121970,
%T A175656 254862,531570,1106830,2301042,4776846,9903218,20505486,42409074,
%U A175656 87614350,180821106,372827022,768023666,1580786574,3251051634
%N A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3*x^2)/(1-3*x+4*x^3).
%C A175656 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 A175656 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 A175656 Vincenzo Librandi, <a href="/A175656/b175656.txt">Table of n, a(n) for n = 0..1000</a>
%H A175656 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,0,-4).
%F A175656 G.f.: (1-3*x^2)/(1 - 3*x + 4*x^3).
%F A175656 a(n) = 3*a(n-1) - 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=6.
%F A175656 a(n) = ((3*n+22)*2^n - 4*(-1)^n)/18.
%p A175656 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 A175656 CoefficientList[Series[(1 - 3 x^2)/(1 - 3 x + 4 x^3), {x, 0, 29}], x] (* _Michael De Vlieger_, Nov 02 2018 *)
%t A175656 LinearRecurrence[{3,0,-4},{1,3,6},30] (* _Harvey P. Dale_, Aug 12 2020 *)
%o A175656 (Magma) [((3*n+22)*2^n-4*(-1)^n)/18: n in [0..40]]; // _Vincenzo Librandi_, Aug 04 2011
%o A175656 (PARI) vector(40, n, n--; ((3*n+22)*2^n - 4*(-1)^n)/18) \\ _G. C. Greubel_, Nov 03 2018
%Y A175656 Cf. A175655 (central square).
%K A175656 nonn,easy
%O A175656 0,2
%A A175656 _Johannes W. Meijer_, Aug 06 2010