# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a179601 Showing 1-1 of 1 %I A179601 #13 Apr 07 2024 09:09:03 %S A179601 1,6,22,108,460,2088,9208,41136,182704,813600,3618784,16104384, %T A179601 71651008,318820992,1418569600,6311953152,28084886272,124963582464, %U A179601 556023840256,2474023050240,11008138832896,48980603529216,217938687588352 %N A179601 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+4*x)/(1 - 2*x - 10*x^2 - 4*x^3). %C A179601 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 king on the eight side and corner squares but on the central square the king goes crazy and turns into a red king, see A179596. %C A179601 The sequence above corresponds to 6 red king vectors, i.e., A[5] vectors, with decimal values 335, 359, 365, 455, 461 and 485. These vectors lead for the corner squares to A179600 and for the side squares to A123347. %H A179601 Index entries for linear recurrences with constant coefficients, signature (2,10,4). %F A179601 G.f.: ( -1-4*x ) / ( (2*x+1)*(2*x^2 + 4*x - 1) ). %F A179601 a(n) = 2*a(n-1) + 10*a(n-2) + 4*a(n-3) with a(0)=1, a(1)=6 and a(2)=22. %F A179601 a(n) = (-2/5)*(-1/2)^(-n) + ((2+3*A)*A^(-n-1) + (2+3*B)*B^(-n-1))/10 with A = (-1+sqrt(6)/2) and B = (-1-sqrt(6)/2). %F A179601 Limit_{k->oo} a(n+k)/a(k) = (-1)^(n+1)*A016116(n+1)/(A041007(n-1)*sqrt(6) - A041006(n-1)) for n => 1. %p A179601 with(LinearAlgebra): nmax:=22; m:=5; A[1]:= [0,1,0,1,1,0,0,0,0]: A[2]:= [1,0,1,1,1,1,0,0,0]: A[3]:= [0,1,0,0,1,1,0,0,0]: A[4]:= [1,1,0,0,1,0,1,1,0]: A[5]:= [1,1,1,0,0,0,1,1,1]: A[6]:= [0,1,1,0,1,0,0,1,1]: A[7]:= [0,0,0,1,1,0,0,1,0]: A[8]:= [0,0,0,1,1,1,1,0,1]: A[9]:= [0,0,0,0,1,1,0,1,0]: A:=Matrix([A[1],A[2],A[3],A[4],A[5],A[6],A[7],A[8],A[9]]): 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); %Y A179601 Cf. A041006, A041007, A123347, A179596, A179597 (central square), A179600. %K A179601 easy,nonn %O A179601 0,2 %A A179601 _Johannes W. Meijer_, Jul 28 2010 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE