%I #22 Apr 07 2024 09:08:53

%S 1,4,10,26,66,166,414,1026,2530,6214,15214,37154,90546,220294,535230,

%T 1298946,3149506,7630726,18476494,44714786,108168210,261575494,

%U 632367774,1528408194,3693378466,8923553734,21557263150,52071634466

%N 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.

%C 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.

%C 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).

%C From _Clark Kimberling_, Aug 23 2017 (Start)

%C p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3.

%C 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.

%C (End)

%H Vincenzo Librandi, <a href="/A175658/b175658.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2).

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

%F 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.

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

%F a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - _Colin Barker_, Aug 29 2017

%p 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);

%t LinearRecurrence[{4,-3,-2},{1,4,10},30] (* _Harvey P. Dale_, Jun 18 2013 *)

%t CoefficientList[Series[(1 - 3 x^2) / (1 - 4 x + 3 x^2 + 2 x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Jul 21 2013 *)

%o (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

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

%Y Cf. A175654, A175655 (central square).

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

%K easy,nonn

%O 0,2

%A _Johannes W. Meijer_, Aug 06 2010