A180036 - OEIS

%I #17 Sep 08 2022 08:45:54

%S 1,3,18,99,549,3042,16857,93411,517626,2868363,15894693,88078554,

%T 488076849,2704619907,14987330082,83050510131,460214540901,

%U 2550224234898,14131764797193,78309496690659,433942777844874

%N Eight white queens and one red queen on a 3 X 3 chessboard. G.f.: (1 - 2*x)/(1 - 5*x - 3*x^2).

%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 white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180028.

%C The sequence above corresponds to 56 red queen vectors, i.e., A[5] vector, with decimal values varying between 7 and 448. The corner and side squares lead for these vectors to A180035.

%H Vincenzo Librandi, <a href="/A180036/b180036.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, 3).

%F G.f.: (1-2*x)/(1 - 5*x - 3*x^2).

%F a(n) = 5*a(n-1) + 3*a(n-2) with a(0) = 1 and a(1) = 3.

%F a(n) = ((1+16*A)*A^(-n-1) + (1+16*B)*B^(-n-1))/37 with A = (-5+sqrt(37))/6 and B = (-5-sqrt(37))/6.

%F a(n) = Sum_{k=0..n} A202395(n,k)*2^k. - _Philippe Deléham_, Dec 21 2011

%p with(LinearAlgebra): nmax:=21; m:=5; A[5]:= [0,0,0,0,0,0,1,1,1]: A:=Matrix([[0,1,1,1,1,0,1,0,1], [1,0,1,1,1,1,0,1,0], [1,1,0,0,1,1,1,0,1], [1,1,0,0,1,1,1,1,0], A[5], [0,1,1,1,1,0,0,1,1], [1,0,1,1,1,0,0,1,1], [0,1,0,1,1,1,1,0,1], [1,0,1,0,1,1,1,1,0]]): 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[{5,3},{1,3},201] (* _Vincenzo Librandi_, Nov 15 2011 *)

%o (Magma) I:=[1,3]; [n le 2 select I[n] else 5*Self(n-1)+3*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011

%K easy,nonn

%O 0,2

%A _Johannes W. Meijer_, Aug 09 2010

%E Second formula corrected by _Vincenzo Librandi_, Nov 15 2011