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A005605 a(n) = a(n-1) + (-1)^(n-1) * a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.

%I A005605 M0176 #37 Jun 18 2025 01:01:26
%S A005605 0,1,1,2,1,5,4,29,13,854,685,730001,260776,532901720777,464897598601,
%T A005605 283984244007552571082330,67854466822576053925129,
%U A005605 80647050844541158378200602801460469923692154029
%N A005605 a(n) = a(n-1) + (-1)^(n-1) * a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.
%D A005605 J. Mestel, Archimedeans Problems Drive 1977, Eureka, 39 (1978), 38-40.
%D A005605 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005605 Reinhard Zumkeller, <a href="/A005605/b005605.txt">Table of n, a(n) for n = 0..25</a>
%H A005605 J. Mestel, <a href="/A005605/a005605.pdf">Archimedeans Problems Drive 1977</a>, Eureka, 39 (1978), 38-40. (Annotated scanned copy)
%F A005605 For odd n, a(n) = floor(c^(2^((n-5)/2))) where c = 5.40649051189479721671529024151000148043062587729806430849172463824709059699198634362266484243.... - _Gerald McGarvey_, Dec 08 2004
%F A005605 The alternating version of this sequence, b(n) = (-1)^(n-1)*a(n) = 0,1,-1,2,-1,5,-4,29,-13,..., satisfies b(n) = b(n-2)^2 - b(n-1). - _Franklin T. Adams-Watters_, Feb 27 2007
%F A005605 a(n+4) = a(n+2) + a(n)^2*(a(n+1) + a(n+2)) for all n>=0. - _Michael Somos_, Jun 16 2025
%t A005605 t = {0, 1}; Do[AppendTo[t, t[[-2]]^2 - t[[-1]]], {n, 18}]; Abs[t] (* _Vladimir Joseph Stephan Orlovsky_, Feb 23 2012 *)
%t A005605 nxt[{n_,a1_,a2_}]:={n+1,a2,a2+(-1)^n*a1^2}; Drop[Transpose[NestList[nxt,{0,0,1},20]][[3]],2] (* _Harvey P. Dale_, Oct 03 2012 *)
%o A005605 (Haskell)
%o A005605 a005605 n = a005605_list !! n
%o A005605 a005605_list = 0 : 1 : zipWith (+) (tail a005605_list)
%o A005605    (zipWith (*) (cycle [-1,1]) (map (^ 2) $ a005605_list))
%o A005605 -- _Reinhard Zumkeller_, Feb 16 2012
%K A005605 nonn,easy,nice
%O A005605 0,4
%A A005605 _N. J. A. Sloane_