A175657 - OEIS

%I #17 Apr 07 2024 09:08:48

%S 1,4,8,18,38,80,166,342,700,1426,2894,5856,11822,23822,47932,96330,

%T 193414,388048,778070,1559334,3123836,6256034,12525598,25073088,

%U 50181598,100420510,200933756,402017562,804277910,1608948656,3218532934

%N 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 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 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 Vincenzo Librandi, <a href="/A175657/b175657.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+x-3*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)=4 and a(2)=8.

%p 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 LinearRecurrence[{3,-1,-2},{1,4,8},40] (* _Harvey P. Dale_, Aug 12 2012 *)

%t 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 (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 Cf. A000045, A000079, A074878, A175654, A175655 (central square).

%K easy,nonn

%O 0,2

%A _Johannes W. Meijer_, Aug 06 2010