A175656 Eight bishops and one elephant on a 3 X 3 chessboard. G.f.: (1-3*x^2)/(1-3*x+4*x^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

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 *)
LinearRecurrence[{3, 0, -4}, {1, 3, 6}, 30] (* Harvey P. Dale, Aug 12 2020 *)

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