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A175657 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 3*2^n - 2*F(n+1), with F(n) = A000045(n).

%I A175657 #17 Apr 07 2024 09:08:48
%S A175657 1,4,8,18,38,80,166,342,700,1426,2894,5856,11822,23822,47932,96330,
%T A175657 193414,388048,778070,1559334,3123836,6256034,12525598,25073088,
%U A175657 50181598,100420510,200933756,402017562,804277910,1608948656,3218532934
%N A175657 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 3*2^n - 2*F(n+1), with F(n) = A000045(n).
%C A175657 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 A175657 The sequence above corresponds to 16 A[5] vectors with decimal values 43, 46, 106, 139, 142, 163, 166, 169, 172, 202, 226, 232, 298, 394, 418 and 424. These vectors lead for the side squares to A000079 and for the corner squares to A074878 (a(n)=3*2^n-2*F(n+2)).
%H A175657 Vincenzo Librandi, <a href="/A175657/b175657.txt">Table of n, a(n) for n = 0..1000</a>
%H A175657 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-2).
%F A175657 G.f.: (1+x-3*x^2)/(1-3*x+x^2+2*x^3).
%F A175657 a(n) = 3*a(n-1)-a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.
%p A175657 with(LinearAlgebra): nmax:=30; m:=5; A[5]:= [0,0,0,1,0,1,0,1,1]: 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 A175657 LinearRecurrence[{3,-1,-2},{1,4,8},40] (* _Harvey P. Dale_, Aug 12 2012 *)
%t A175657 CoefficientList[Series[(1 + x - 3 x^2) / (1 - 3 x + x^2 + 2 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 21 2013 *)
%o A175657 (Magma) I:=[1,4,8]; [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 A175657 Cf. A000045, A000079, A074878, A175654, A175655 (central square).
%K A175657 easy,nonn
%O A175657 0,2
%A A175657 _Johannes W. Meijer_, Aug 06 2010