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.

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

Offset: 0

Author

Johannes W. Meijer, Aug 06 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 24 A[5] 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).

Crossrefs

Cf. A175654, A175655 (central square).

Cf. A000129 (Pell(n)), A078057 (Pell(n)+Pell(n+1)), A094723 (Pell(n+2)-2^n).

Programs

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

Maple

nmax:=27; m:=5; A[5]:= [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[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[{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 *)

PARI

Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017

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_{n->oo} a(n+1)/a(n) = 1+sqrt(2).

a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - Colin Barker, Aug 29 2017