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 A179604 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1+x)/(1-2*x-9*x^2-2*x^3). 2
 1, 3, 15, 59, 259, 1079, 4607, 19443, 82507, 349215, 1479879, 6267707, 26552755, 112474631, 476459471, 2018296131, 8549676763, 36216937647, 153417558423, 649886909195, 2752965719491, 11661748738583, 49399962770975 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 central square the king goes crazy and turns into a red king, see A179596. The sequence above corresponds to 4 red king vectors, i.e. A[5] vectors, with decimal [binary] values 327 [1,0,1,0,0,0,1,1,1], 333 [1,0,1,0,0,1,1,0,1], 357 [1,0,1,1,0,0,1,0,1] and 453 [1,1,1,0,0,0,1,0,1]. These vectors lead for the side squares to A015448 and for the central square to A179605. LINKS Index to sequences with linear recurrences with constant coefficients, signature (2,9,2). FORMULA GF(x) = ( -1-x ) / ( (2*x+1)*(x^2+4*x-1) ). a(n) = 2*a(n-1)+9*a(n-2)+2*a(n-3) with a(0)=1, a(1)=3 and a(2)=15. a(n) = (20*(-1/2)^(-n)+(5+7*sqrt(5))*A^(-n-1)+(5-7*sqrt(5))*B^(-n-1))/110 with A = (-2+sqrt(5)) and B:= (-2-sqrt(5)). Limit(a(n+k)/a(k), k=infinity) = (-1)^(n+1)/(A001076(n)*sqrt(5)-A001077(n)) MAPLE with(LinearAlgebra): 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, 0, 1, 0, 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); MATHEMATICA LinearRecurrence[{2, 9, 2}, {1, 3, 15}, 30] (* or *) CoefficientList[ Series[ (x+1)/(-2 x^3-9 x^2-2 x+1), {x, 0, 30}], x] (* From Harvey P. Dale, Mar 17 2012 *) CROSSREFS Sequence in context: A062473 A218200 A069009 * A176311 A036750 A058748 Adjacent sequences:  A179601 A179602 A179603 * A179605 A179606 A179607 KEYWORD easy,nonn AUTHOR Johannes W. Meijer, Jul 28 2010 STATUS approved

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