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%I A175658 #22 Apr 07 2024 09:08:53
%S A175658 1,4,10,26,66,166,414,1026,2530,6214,15214,37154,90546,220294,535230,
%T A175658 1298946,3149506,7630726,18476494,44714786,108168210,261575494,
%U A175658 632367774,1528408194,3693378466,8923553734,21557263150,52071634466
%N 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.
%C A175658 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 A175658 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 A175658 From _Clark Kimberling_, Aug 23 2017 (Start)
%C A175658 p-INVERT of (1,1,1,....), where p(S) = 1-S-2*S^2+2*S^3.
%C A175658 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 A175658 (End)
%H A175658 Vincenzo Librandi, Table of n, a(n) for n = 0..1000
%H A175658 Index entries for linear recurrences with constant coefficients, signature (4,-3,-2).
%F A175658 G.f.: ( 1-3*x^2 ) / ( (2*x-1)*(x^2+2*x-1) ).
%F A175658 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 A175658 Limit_{n->oo} a(n+1)/a(n) = 1+sqrt(2).
%F A175658 a(n) = (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n) - 2^n. - _Colin Barker_, Aug 29 2017
%p A175658 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 A175658 LinearRecurrence[{4,-3,-2},{1,4,10},30] (* _Harvey P. Dale_, Jun 18 2013 *)
%t A175658 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 A175658 (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 A175658 (PARI) Vec((1 - 3*x^2) / ((1 - 2*x)*(1 - 2*x - x^2)) + O(x^30)) \\ Colin Barker, Aug 29 2017
%Y A175658 Cf. A175654, A175655 (central square).
%Y A175658 Cf. A000129 (Pell(n), A078057 (Pell(n)+Pell(n+1)), A094723 (Pell(n+2)-2^n).
%K A175658 easy,nonn
%O A175658 0,2
%A A175658 _Johannes W. Meijer_, Aug 06 2010
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