A175659 - OEIS

%I #18 Apr 07 2024 09:08:58

%S 1,6,16,52,156,476,1436,4332,13036,39196,117756,353612,1061516,

%T 3185916,9560476,28686892,86071596,258236636,774753596,2324348172,

%U 6973219276,20920007356,62760721116,188283561452,564853480556

%N Eight bishops and one elephant on a 3 X 3 chessboard: a(n)= (3^(n+1)-Jacobsthal(n+1))-(3^n-Jacobsthal(n)), with Jacobsthal=A001045.

%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 4 A[5] vectors with decimal values 343, 349, 373 and 469. These vectors lead for the side squares to A000079 and for the corner squares to A093833 (a(n)=3^n-Jacobsthal(n)).

%H Vincenzo Librandi, <a href="/A175659/b175659.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 (4,-1,-6).

%F G.f.: (1+2*x-7*x^2)/(1-4*x+x^2+6*x^3).

%F a(n) = 4*a(n-1)-a(n-2)-6*a(n-3) with a(0)=1, a(1)=6 and a(2)=16.

%F a(n) = (-2*(-1)^n)/3-2^n/3+2*3^n. [_Colin Barker_, Oct 07 2012]

%p nmax:=24; m:=5; A[5]:= [1,0,1,0,1,0,1,1,1]: 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 + 2 x - 7 x^2) / (1 - 4 x + x^2 + 6 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 21 2013 *)

%o (Magma) I:=[1, 6, 16]; [n le 3 select I[n] else 4*Self(n-1)-Self(n-2)-6*Self(n-3): n in [1..35]]; // _Vincenzo Librandi_, Jul 21 2013

%Y Cf. A000079, A001045, A093833, A175654, A175655 (central square).

%K easy,nonn

%O 0,2

%A _Johannes W. Meijer_, Aug 06 2010