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 A175655 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3). 24
 1, 4, 8, 22, 50, 124, 290, 694, 1628, 3838, 8978, 21004, 48962, 114022, 265004, 615262, 1426658, 3305212, 7650722, 17697430, 40911740, 94528318, 218312114, 503994220, 1163124866, 2683496134, 6189647948, 14273690782 (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. For the central square the 512 elephants lead to 46 different elephant sequences, see the cross-references for examples. The sequence above corresponds to 16 A[5] vectors with decimal values 71, 77, 101, 197, 263, 269, 293, 323, 326, 329, 332, 353, 356, 389, 449 and 452. These vectors lead for the side squares to A000079 and for the corner squares to A175654. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (3,1,-6). FORMULA G.f.: (1+x-5*x^2)/(1-3*x-x^2+6*x^3). a(n) = 3*a(n-1)+a(n-2)-6*a(n-3) with a(0)=1, a(1)=4 and a(2)=8. a(n) = ((10+8*A)*A^(-n-1)+(10+8*B)*B^(-n-1))/13-2^n with A = (-1+sqrt(13))/6 and B = (-1-sqrt(13))/6. Limit(a(n+k)/a(k), k=infinity) = (-1)^(n)*2*A000244(n)/(A075118(n)-A006130(n-1)*sqrt(13)). MAPLE with(LinearAlgebra): nmax:=27; m:=5; A[5]:= [0, 0, 1, 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 + x - 5 x^2) / (1 - 3 x - x^2 + 6 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *) PROG (MAGMA) I:=[1, 4, 8]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-6*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013 (PARI) a(n)=([0, 1, 0; 0, 0, 1; -6, 1, 3]^n*[1; 4; 8])[1, 1] \\ Charles R Greathouse IV, Oct 03 2016 CROSSREFS Cf. Elephant sequences central square [decimal value A[5]]: A000007 [0], A000012 [16], A000045 [1], A011782 [2], A000079 [3], A003945 [42], A099036 [11], A175656 [7], A105476 [69], A168604 [26], A045891 [19], A078057 [21], A151821 [170], A175657 [43], 4*A172481 [15; n>=-1], A175655 [71, this sequence], 4*A026597 [325; n>=-1], A033484 [58], A087447 [27], A175658 [23], A026150 [85], A175661 [171], A036563 [186], A098156 [59], A046717 [341], 2*A001792 [187; n>=1 with a(0)=1], A175659 [343]. Sequence in context: A003606 A048657 A322284 * A000639 A190795 A052528 Adjacent sequences:  A175652 A175653 A175654 * A175656 A175657 A175658 KEYWORD easy,nonn AUTHOR Johannes W. Meijer, Aug 06 2010, Aug 10 2010 STATUS approved

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Last modified December 15 04:20 EST 2019. Contains 329991 sequences. (Running on oeis4.)