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 A175658 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2*Pell(n+1)+2*Pell(n)-2^n, with Pell = A000129. 4
 1, 4, 10, 26, 66, 166, 414, 1026, 2530, 6214, 15214, 37154, 90546, 220294, 535230, 1298946, 3149506, 7630726, 18476494, 44714786, 108168210, 261575494, 632367774, 1528408194, 3693378466, 8923553734, 21557263150, 52071634466 (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 vectors with decimal values 23, 29, 53, 83, 86, 89, 92, 113, 116, 149, 209, 212, 275, 278, 281, 284, 305, 308, 338, 344, 368, 401, 404 and 464. These vectors lead for the side squares to A000079 and for the corner squares to 2*A094723 (a(n)=2*Pell(n+1)-2^n). From Clark Kimberling, Aug 23 2017 (Start) p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3. Suppose s = (c(0), c(1), c(2),...) is a sequence and p(S) is a polynomial. Let S(x) = c(0)*x + c(1)*x^2 + c(2)*x^3 + ... and T(x) = (-p(0) + 1/p(S(x)))/x. The p-INVERT of s is the sequence t(s) of coefficients in the Maclaurin series for T(x).  Taking p(S) = 1 - S gives the "INVERT" transform of s, so that p-INVERT is a generalization of the "INVERT" transform (e.g., A033453). See A291000 for a guide to related sequences. (End) LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (4,-3,-2). FORMULA G.f.: ( 1-3*x^2 ) / ( (2*x-1)*(x^2+2*x-1) ). a(n) = 4*a(n-1)-3*a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=10. Limit(a(n+1)/a(n), n=infinity) = 1+sqrt(2). a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - Colin Barker, Aug 29 2017 MAPLE nmax:=27; m:=5; A:= [0, 0, 0, 0, 1, 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, [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 LinearRecurrence[{4, -3, -2}, {1, 4, 10}, 30] (* Harvey P. Dale, Jun 18 2013 *) CoefficientList[Series[(1 - 3 x^2) / (1 - 4 x + 3 x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *) PROG (MAGMA) I:=[1, 4, 10]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 21 2013 (PARI) Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017 CROSSREFS Cf. A175655 (central square). Cf. A000129 (Pell(n), A078057 (Pell(n)+Pell(n+1)), A094723 (Pell(n+2)-2^n). Sequence in context: A133086 A285186 A178037 * A191605 A277236 A218208 Adjacent sequences:  A175655 A175656 A175657 * A175659 A175660 A175661 KEYWORD easy,nonn AUTHOR Johannes W. Meijer, Aug 06 2010 STATUS approved

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Last modified January 21 21:30 EST 2020. Contains 331128 sequences. (Running on oeis4.)