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A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 11*x^2 - 6*x^3).

%I A179596 #32 Jul 10 2022 09:42:48
%S A179596 1,3,17,73,351,1607,7513,34809,161903,751783,3493353,16227737,
%T A179596 75393055,350251335,1627192697,7559508409,35119644495,163157037671,
%U A179596 757987215241,3521419711833,16359641017343,76002822156295,353090213774361
%N A179596 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + x)/(1 - 2*x - 11*x^2 - 6*x^3).
%C A179596 The a(n) represent the number of n-move routes of a fairy chess piece starting in a given corner square (m = 1, 3, 7 or 9) on a 3 X 3 chessboard. This fairy chess piece behaves like a king on the eight side and corner squares but on the center square the king goes crazy and turns into a red king.
%C A179596 On a 3 X 3 chessboard there are 2^9 = 512 ways to go crazy on the center square (off the center the piece behaves like a normal king). The red king is represented by the A[5] vector in the fifth row of the adjacency matrix A, see the Maple program and A180140. For the corner squares the 512 red kings lead to 47 different red king sequences, see the overview of the red king sequences.
%C A179596 The sequence above corresponds to four A[5] vectors with decimal [binary] values 367 [101 101 111], 463 [111 001 111], 487 [111 100 111] and 493 [111 101 101]. These vectors lead for the side squares to A126473 and for the central square to A179597.
%C A179596 This sequence belongs to a family of sequences with g.f. (1+x)/(1 - 2*x - (k+8)*x^2 - 2*k*x^3). Red king sequences that are members of this family are A083424 (k=0), A179604 (k=1), A179600 (k=2), A179596 (k=3; this sequence) and A086346 (k=4). Other members of this family are A015528 (k=5) and A179608 (k=-4).
%D A179596 Gary Chartrand, Introductory Graph Theory, pp. 217-221, 1984.
%H A179596 Vincenzo Librandi, <a href="/A179596/b179596.txt">Table of n, a(n) for n = 0..100</a>
%H A179596 Charles Krauthammer, <a href="http://www.time.com/time/columnist/krauthammer/article/0,9565,1054411,00.html">Did Chess Make Him Crazy?</a>, Time, April 26, 2005.
%H A179596 Johannes W. Meijer, <a href="/A179596/a179596.jpg">The red king sequences</a>.
%H A179596 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,11,6).
%F A179596 G.f.: (1+x)/(1 - 2*x - 11*x^2 - 6*x^3).
%F A179596 a(n) = 2*a(n-1) + 11*a(n-2) + 6*a(n-3) with a(0)=1, a(1)=3 and a(2)=17.
%F A179596 a(n) = (-1)^(-n)*2^(n+1)/9 + ((49+17*sqrt(7))*A^(-n) + (49-17*sqrt(7))*B^(-n))/126 with A = (-2+sqrt(7))/3 and B = (-2-sqrt(7))/3.
%F A179596 Lim_{k->infinity} a(n+k)/a(k) = (-1)^(n+1)*A000244(n)/(A015530(n)*sqrt(7) - A108851(n)).
%p A179596 nmax:=22; m:=1; 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]:= [1,0,1,1,0,1,1,1,1]: 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);
%t A179596 LinearRecurrence[{2,11,6},{1,3,17},30] (* _Harvey P. Dale_, May 18 2011 *)
%o A179596 (PARI) Vec((1+x)/(1-2*x-11*x^2-6*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Jul 16 2011
%Y A179596 Cf. A180140 (berserker sequences).
%Y A179596 Cf. Red king sequences corner squares [decimal value A[5]]: A086346 [495], A015525 [239], A179596 [367], A179600 [335], A015524 [95], A083858 [31], A179604 [327], A015523 [27], A179610 [85], A083424 [325], A015521 [11], A007482 [2], A014335 [16].
%K A179596 easy,nonn
%O A179596 0,2
%A A179596 _Johannes W. Meijer_, Jul 28 2010; edited Jun 21 2013