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%I #12 Apr 28 2018 12:09:43
%S 1,1,1,0,2,-3,8,-15,35,-63,146,-264,611,-1104,2555,-4617,10685,-19308,
%T 44684,-80745,186866,-337671,781463,-1412121,3268034,-5905410,
%U 13666733,-24696090,57153503,-103277649,239012711,-431901276,999537614,-1806186663,4180009664,-7553370279
%N a(1)=a(2)=a(3)=1; a(n)=|a(n-2)+a(n-3)|-2*a(n-1).
%F For n>4, a(2n)+2a(2n-1)+a(2n-2)+a(2n-3)=0.
%F Let c=4.181943336... be the largest positive root of 1-x-4*x^2+x^3. Then a(2n) is asymptotic to a*c^n, where a=-0.0493553440330328..., while a(2n+1) is asymptotic to b*c^n, where b= 0.11422175750398... (-a/b is the smallest root (in absolute value) of 3-12*x+13*x^2-3*x^3.)
%F limit n ->infinity a(2n+1)/a(2n)=-2.3142733525986128..., the largest root (in absolute value) of 3+13*x+12*x^2+3*x^3; limit n ->infinity a(2n)/a(2n-1)=-1.8070222... the largest root (in absolute value) of -3+3*x+8*x^2+3*x^3.
%F Empirical G.f.: x*(2*x^8-2*x^7-4*x^5-3*x^4-4*x^3-3*x^2+x+1) / (x^6-x^4-4*x^2+1). [_Colin Barker_, Dec 01 2012]
%t nxt[{a_,b_,c_}]:={b,c,Abs[a+b]-2c}; NestList[nxt,{1,1,1},40][[All,1]] (* _Harvey P. Dale_, Apr 28 2018 *)
%K sign,easy
%O 1,5
%A _Benoit Cloitre_, Feb 05 2003
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