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 A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3*x^2)/(1-3*x+4*x^3). 3
 1, 3, 6, 14, 30, 66, 142, 306, 654, 1394, 2958, 6258, 13198, 27762, 58254, 121970, 254862, 531570, 1106830, 2301042, 4776846, 9903218, 20505486, 42409074, 87614350, 180821106, 372827022, 768023666, 1580786574, 3251051634 (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 the central square (m = 5) on a 3 X 3 chessboard. This fairy chess piece behaves like a bishop on the eight side and corner squares but on the central square the bishop turns into a raging elephant, see A175654. The sequence above corresponds to 24 A[5] vectors with decimal values 7, 13, 37, 67, 70, 73, 76, 97, 100, 133, 193, 196, 259, 262, 265, 268, 289, 292, 322, 328, 352, 385, 388 and 448. These vectors lead for the side squares to A000079 and for the corner squares to A172481. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,0,-4). FORMULA G.f.: (1-3*x^2)/(1 - 3*x + 4*x^3). a(n) = 3*a(n-1) - 4*a(n-3) with a(0)=1, a(1)=3 and a(2)=6. a(n) = ((3*n+22)*2^n - 4*(-1)^n)/18. MAPLE with(LinearAlgebra): nmax:=29; m:=5; A[5]:= [0, 0, 0, 0, 0, 0, 1, 1, 1]: A:=Matrix([[0, 0, 0, 0, 1, 0, 0, 0, 1], [0, 0, 0, 1, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0, 1, 0, 0], [0, 1, 0, 0, 0, 0, 0, 1, 0], A[5], [0, 1, 0, 0, 0, 0, 0, 1, 0], [0, 0, 1, 0, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 1, 0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 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 CoefficientList[Series[(1 - 3 x^2)/(1 - 3 x + 4 x^3), {x, 0, 29}], x] (* Michael De Vlieger, Nov 02 2018 *) PROG (MAGMA) [((3*n+22)*2^n-4*(-1)^n)/18: n in [0..40]]; // Vincenzo Librandi, Aug 04 2011 (PARI) vector(40, n, n--; ((3*n+22)*2^n - 4*(-1)^n)/18) \\ G. C. Greubel, Nov 03 2018 CROSSREFS Cf. A175655 (central square). Sequence in context: A083797 A308580 A192672 * A196450 A131244 A077926 Adjacent sequences:  A175653 A175654 A175655 * A175657 A175658 A175659 KEYWORD nonn,easy AUTHOR Johannes W. Meijer, Aug 06 2010 STATUS approved

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Last modified January 19 06:37 EST 2020. Contains 331033 sequences. (Running on oeis4.)