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 A179602 Eight white kings and one red king on a 3 X 3 chessboard. G.f.: (1 + 2*x)/(1 - 3*x - 7*x^2). 3
 1, 5, 22, 101, 457, 2078, 9433, 42845, 194566, 883613, 4012801, 18223694, 82760689, 375847925, 1706868598, 7751541269, 35202703993, 159868900862, 726025630537, 3297159197645, 14973657006694, 68001085403597, 308818855257649 (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 side square (m = 2, 4, 6 or 8) 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 24 red king vectors, i.e., A[5] vectors, with decimal values 95, 119, 125, 215, 221, 245, 287, 311, 317, 347, 350, 371, 374, 377, 380, 407, 413, 437, 467, 470, 473, 476, 497 and 500. These vectors lead for the corner squares to A015524 and for the central square to A179603. LINKS Index entries for linear recurrences with constant coefficients, signature (3, 7). FORMULA G.f.: (1+2*x)/(1 - 3*x - 7*x^2). a(n) = 3*a(n-1) + 7*a(n-2) with a(0) = 1 and a(1) = 5. a(n) = ((37+4*37^(1/2))*A^(-n-1) + (37-4*37^(1/2))*B^(-n-1))/259 with A = (-3+sqrt(37))/14 and B = (-3-sqrt(37))/14. MAPLE with(LinearAlgebra): nmax:=22; m:=2; 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, 1, 0, 0, 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); MATHEMATICA CoefficientList[Series[(1+2x)/(1-3x-7x^2), {x, 0, 40}], x] (* or *) LinearRecurrence[ {3, 7}, {1, 5}, 40] (* Harvey P. Dale, Mar 28 2013 *) CROSSREFS Cf. A126473 (side squares). Sequence in context: A087439 A033452 A295519 * A262440 A296044 A048251 Adjacent sequences:  A179599 A179600 A179601 * A179603 A179604 A179605 KEYWORD easy,nonn AUTHOR Johannes W. Meijer, Jul 28 2010 STATUS approved

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Last modified February 19 00:35 EST 2020. Contains 332028 sequences. (Running on oeis4.)