A175657 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 3*2^n - 2*F(n+1), with F(n) = A000045(n).

1, 4, 8, 18, 38, 80, 166, 342, 700, 1426, 2894, 5856, 11822, 23822, 47932, 96330, 193414, 388048, 778070, 1559334, 3123836, 6256034, 12525598, 25073088, 50181598, 100420510, 200933756, 402017562, 804277910, 1608948656, 3218532934

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 16 A[5] vectors with decimal values 43, 46, 106, 139, 142, 163, 166, 169, 172, 202, 226, 232, 298, 394, 418 and 424. These vectors lead for the side squares to A000079 and for the corner squares to A074878 (a(n)=3*2^n-2*F(n+2)).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (3,-1,-2).

Formula

G.f.: (1+x-3*x^2)/(1-3*x+x^2+2*x^3).

a(n) = 3*a(n-1)-a(n-2)-2*a(n-3) with a(0)=1, a(1)=4 and a(2)=8.

Maple

with(LinearAlgebra): nmax:=30; m:=5; A[5]:= [0, 0, 0, 1, 0, 1, 0, 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

LinearRecurrence[{3, -1, -2}, {1, 4, 8}, 40] (* Harvey P. Dale, Aug 12 2012 *)
CoefficientList[Series[(1 + x - 3 x^2) / (1 - 3 x + x^2 + 2 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)-2*Self(n-3): n in [1..35]]; // Vincenzo Librandi, Jul 21 2013

Crossrefs

Cf. A000045, A000079, A074878, A175654, A175655 (central square).

Sequence in context: A008204 A190062 A228231 * A080287 A280155 A075310

Adjacent sequences: A175654 A175655 A175656 * A175658 A175659 A175660

Keyword

easy,nonn

Author

Johannes W. Meijer, Aug 06 2010

Status

approved