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%I #56 Sep 08 2022 08:45:18
%S 1,1,-2,-5,1,16,13,-35,-74,31,253,160,-599,-1079,718,3955,1801,-10064,
%T -15467,14725,61126,16951,-166427,-217280,282001,933841,87838,
%U -2713685,-2977199,5163856,14095453,-1396115,-43682474,-39494129,91553293,210035680,-64624199,-694731239,-500858642
%N Expansion of 1/(1-x*(1-3*x)).
%C Row sums of Riordan array (1, x*(1-3*x)). In general, Sum_{k=0..n} (-1)^(n-k)*binomial(k,n-k)*r^(n-k) yields the row sums of the Riordan array (1, x(1-kx)).
%C Row sums of Riordan array (1/(1+3*x^2), x/(1+3*x^2)). - _Paul Barry_, Sep 10 2005
%C See A214733 for a differently signed version of this sequence. - _Peter Bala_, Nov 21 2016
%H Vincenzo Librandi, <a href="/A106852/b106852.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,-3).
%F From _Paul Barry_, Sep 10 2005: (Start)
%F G.f.: 1/(1-x+3*x^2).
%F a(n) = 2*sqrt(33)*3^(n/2)*cos((n+1)*arctan(sqrt(11)/11)-pi*n/2)/11.
%F a(n) = 3^(n/2)(cos(-n*arccot(sqrt(11)/11))-sqrt(11)*sin(-n*arccot(sqrt(11)/11))/11).
%F a(n) = ((1+sqrt(-11))^(n+1)-(1-sqrt(-11))^(n+1))/(2^(n+1)sqrt(-11)).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(k, n-k)*3^(n-k) = Sum_{k=0..n} A109466(n,k)*3^(n-k).
%F a(n) = Sum_{k=0..n} C((n+k)/2, k)*(-3)^((n-k)/2)*(1+(-1)^(n-k))/2.
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)(-3)^k. (End)
%F a(n) = a(n-1) - 3*a(n-2), a(0)=1, a(1)=1. - _Philippe Deléham_, Oct 21 2008
%F G.f.: Q(0)/x -1/x, where Q(k) = 1 - 3*x^2 + (k+2)*x - x*(k+1 - 3*x)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 07 2013
%t CoefficientList[Series[1/(1 - x (1 - 3 x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 07 2013 *)
%t LinearRecurrence[{1,-3},{1,1},40] (* _Harvey P. Dale_, Apr 02 2016 *)
%o (Sage) [lucas_number1(n,1,+3) for n in range(1, 40)] # _Zerinvary Lajos_, Apr 22 2009
%o (PARI) a(n)=([0,1; -3,1]^n*[1;1])[1,1] \\ _Charles R Greathouse IV_, Nov 21 2016
%o (PARI) x='x+O('x^30); Vec(1/(1-x+3*x^2)) \\ _G. C. Greubel_, Jan 14 2018
%o (Magma) I:=[1,1]; [n le 2 select I[n] else Self(n-1) - 3*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Jan 14 2018
%Y Cf. A214733.
%K sign,easy
%O 0,3
%A _Paul Barry_, May 08 2005
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