A003687 a(n+1) = a(n)-a(1)a(2)...a(n-1), if n>0. a(0)=1, a(1)=2.

1, 2, 1, -1, -3, -1, -7, -1, -43, -1, -1807, -1, -3263443, -1, -10650056950807, -1, -113423713055421844361000443, -1, -12864938683278671740537145998360961546653259485195807, -1

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign
• easy

Comments

a(n) = a(n-1)-a(n-2)^2+a(n-1)*a(n-2), if n>2. - Michael Somos, Mar 19 2004

Consider the mapping f(a/b) = (a - b)/(ab). Taking a = 2 b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 2/1,1/2,-1/2,-3/-2,-1/6,... Sequence contains the numerators. - Amarnath Murthy, Mar 24 2003

An infinite coprime sequence defined by recursion. - Michael Somos, Mar 19 2004

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..27

Crossrefs

Cf. A081478.

For n>1, a(2n-1) = -1, a(2n) = -A007018(n-1) - 1.

Programs

Magma

I:=[1,2,1]; [n le 3 select I[n] else Self(n-1)-Self(n-2)^2+Self(n-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

Mathematica

{1}~Join~NestList[{(#1 - #2), #1 #2} & @@ # &, {2, 1}, 17] [[All, 1]] (* Michael De Vlieger, Sep 04 2016 *)

PARI

a(n)=local(an); if(n<1,(n==0),an=vector(max(2,n)); an[1]=2; an[2]=1; for(k=3,n,an[k]=an[k-1]-an[k-2]^2+an[k-1]*an[k-2]); an[n])

Sage

def A003687():
    x, y = 2, 1
    yield y
    while true:
       yield x
       x, y = x - y, x * y
a = A003687(); print([next(a) for i in range(20)])  # Peter Luschny, Dec 17 2015