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

0, 1, 1, 2, 1, 5, 4, 29, 13, 854, 685, 730001, 260776, 532901720777, 464897598601, 283984244007552571082330, 67854466822576053925129, 80647050844541158378200602801460469923692154029

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• J. Mestel, Archimedeans Problems Drive 1977, Eureka, 39 (1978), 38-40.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..25
• J. Mestel, Archimedeans Problems Drive 1977, Eureka, 39 (1978), 38-40. (Annotated scanned copy)

Programs

Haskell

a005605 n = a005605_list !! n
a005605_list = 0 : 1 : zipWith (+) (tail a005605_list)
   (zipWith (*) (cycle [-1,1]) (map (^ 2) $ a005605_list))
-- Reinhard Zumkeller, Feb 16 2012

Mathematica

t = {0, 1}; Do[AppendTo[t, t[[-2]]^2 - t[[-1]]], {n, 18}]; Abs[t] (* Vladimir Joseph Stephan Orlovsky, Feb 23 2012 *)
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 *)

Formula

For odd n, a(n) = floor(c^(2^((n-5)/2))) where c = 5.40649051189479721671529024151000148043062587729806430849172463824709059699198634362266484243.... - Gerald McGarvey, Dec 08 2004

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

a(n+4) = a(n+2) + a(n)^2*(a(n+1) + a(n+2)) for all n>=0. - Michael Somos, Jun 16 2025