A175661 Eight bishops and one elephant on a 3 X 3 chessboard: a(n) = 2^(n+2)-3*F(n+1), with F(n) = A000045(n).

1, 5, 10, 23, 49, 104, 217, 449, 922, 1883, 3829, 7760, 15685, 31637, 63706, 128111, 257353, 516536, 1036033, 2076857, 4161466, 8335475, 16691245, 33415328, 66883789, 133853549, 267846202, 535917479, 1072199137, 2144987528

Offset: 0

Author

Johannes W. Meijer, Aug 06 2010, Aug 10 2010

Keywords

• easy
• nonn

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 four A[5] vectors with decimal values 171, 174, 234 and 426. These vectors lead for the side squares to A000079 and for the corner squares to A175660 (a(n)=2^(n+2)-3*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).

Crossrefs

Cf. A175655 (central square), A000045.

Cf. A027973 (2^(n+2)+F(n)-F(n+4)), A099036 (2^n-F(n)), A167821 (2^(n+1)-2*F(n+2)), A175657 (3*2^n-2*F(n+1)), A175660 (2^(n+2)-3*F(n+2)), A179610 (convolution of (-4)^n and F(n+1)).

Programs

Magma

I:=[1,5,10]; [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

Maple

nmax:=29; m:=5; A[5]:= [0,1,0,1,0,1,0,1,1]: 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

CoefficientList[Series[(1 + 2 x - 4 x^2) / (1 - 3 x + x^2 + 2 x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 21 2013 *)
LinearRecurrence[{3,-1,-2},{1,5,10},30] (* Harvey P. Dale, Apr 15 2019 *)

Formula

G.f.: (1 + 2*x - 4*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)=5 and a(2)=10.