cp's OEIS Frontend

A180030 Number of n-move paths on a 3 X 3 chessboard of a queen starting or ending in a corner or side square.

%I A180030 #12 Sep 08 2022 08:45:54
%S A180030 1,6,38,238,1494,9374,58822,369102,2316086,14533246,91194918,
%T A180030 572240558,3590762134,22531735134,141384772742,887177744782,
%U A180030 5566966905846,34932256487486,219197017684198,1375443140320878,8630791843077974
%N A180030 Number of n-move paths on a 3 X 3 chessboard of a queen starting or ending in a corner or side square.
%C A180030 The a(n) represent the number of n-move paths of a chess queen starting or ending in a given corner or side square (m = 1, 3, 7, 9; 2, 4, 6, 8) on a 3 X 3 chessboard. The central square leads to A180031.
%C A180030 To determine the a(n) we can either sum the components of the column vector A^n[k,m], with A the adjacency matrix of the queen's graph, or we can sum the components of the row vector A^n[m,k], see the Maple program.
%C A180030 Closely related with this sequence are the red queen sequences, see A180028 and A180032.
%C A180030 Inverse binomial transform of A015555 (without the leading 0).
%H A180030 Vincenzo Librandi, <a href="/A180030/b180030.txt">Table of n, a(n) for n = 0..200</a>
%H A180030 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5, 8).
%F A180030 G.f.: (1+x)/(1 - 5*x - 8*x^2).
%F A180030 a(n) = 5*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 6.
%F A180030 a(n) = ((7+11*A)*A^(-n-1) + (7+11*B)*B^(-n-1))/57 with A = (-5+sqrt(57))/16 and B = (-5-sqrt(57))/16.
%p A180030 with(LinearAlgebra): nmax:=20; m:=1; A[5]:= [1,1,1,1,0,1,1,1,1]: 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 A180030 LinearRecurrence[{5,8},{1,6},201] (* _Vincenzo Librandi_, Nov 15 2011 *)
%o A180030 (Magma) I:=[1,6]; [n le 2 select I[n] else 5*Self(n-1)+8*Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Nov 15 2011
%K A180030 nonn,easy
%O A180030 0,2
%A A180030 _Johannes W. Meijer_, Aug 09 2010