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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).
4
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)).
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
KEYWORD
easy,nonn
AUTHOR
Johannes W. Meijer, Aug 06 2010
STATUS
approved