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%I #33 Sep 08 2022 08:45:11
%S 0,1,2,-2,-16,-20,56,232,128,-1136,-3040,736,19712,35008,-48256,
%T -306560,-323584,1192192,4325888,1498624,-22958080,-54907904,27932672,
%U 385312768,603029504,-1105817600,-5829812224,-5024718848,24929435648,80007184384
%N a(n) = 2*a(n-1) - 6*a(n-2), a(0)=0, a(1)=1.
%H G. C. Greubel, <a href="/A088139/b088139.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 (2,-6).
%F G.f.: x/(1-2*x+6*x^2).
%F E.g.f.: exp(x)*sin(sqrt(5)*x)/sqrt(5).
%F a(n) = ((1+i*sqrt(5))^n-(1-i*sqrt(5))^n)/(2*i*sqrt(5)).
%F a(n) = Im{(1+i*sqrt(5))^n/sqrt(5)}.
%F a(n) = Sum_{k=0..floor(n/2)} C(n, 2k+1)(-5)^k.
%F a(n+1) = (-1)^n*Sum_{k, 0<=k<=n} A172250(n,k)*(-2)^k. - _Philippe Deléham_, Feb 15 2012
%p seq(coeff(series(x/(1-2*x+6*x^2),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 23 2018
%t Join[{a=0,b=1},Table[c=2*b-6*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 17 2011*)
%t TrigExpand@Table[(6^(n/2) Sin[n ArcTan[Sqrt[5]]])/Sqrt[5], {n, 0, 20}] (* or *)
%t Table[Sum[(-5)^k Binomial[n, 2 k + 1], {k, 0, n/2}], {n, 0, 20}] (* _Vladimir Reshetnikov_, Sep 20 2016 *)
%o (Sage) [lucas_number1(n,2,6) for n in range(0, 30)] # _Zerinvary Lajos_, Apr 23 2009
%o (PARI) x='x+O('x^30); concat([0], Vec(x/(1-2*x+6*x^2))) \\ _G. C. Greubel_, Oct 22 2018
%o (Magma) I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Oct 22 2018
%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-6*a[n-2]; od; a; # _Muniru A Asiru_, Oct 23 2018
%Y Cf. A045873, A084102, A088136, A088137.
%K easy,sign
%O 0,3
%A _Paul Barry_, Sep 20 2003
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