OFFSET
0,2
COMMENTS
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.
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.
Inverse binomial transform of A015555 (without the leading 0).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (5, 8).
FORMULA
G.f.: (1+x)/(1 - 5*x - 8*x^2).
a(n) = 5*a(n-1) + 8*a(n-2) with a(0) = 1 and a(1) = 6.
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.
MAPLE
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);
MATHEMATICA
LinearRecurrence[{5, 8}, {1, 6}, 201] (* Vincenzo Librandi, Nov 15 2011 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Johannes W. Meijer, Aug 09 2010
STATUS
approved