A144743 Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=3.

3, 5, 19, 341, 115939, 13441735781, 180680260792773944179, 32645356640144805339284259388335434039861, 1065719310162246533488642668727242229836148490441005113524301742665845135502859459

Offset: 0

Author

Artur Jasinski, Sep 20 2008

Keywords

• nonn

Comments

a(0)=3 is the smallest integer generating an increasing sequence of the form a(n)=a(n-1)^2-a(n-1)-1.

Conjecture: A130282 and this sequence are disjoint. If this is true, for n >= 1, a(n+1) is the smallest m such that (m^2-1) / (a(n)^2-1) + 1 is a square. - Jianing Song, Mar 19 2022

Crossrefs

Cf. A000058, A082732, A144744, A144745, A144746, A144747, A144748.

Programs

Mathematica

a = {3}; k = 3; Do[k = k^2 - k - 1; AppendTo[a, k], {n, 1, 10}]; a

PARI

a(n,s=3)=for(i=1,n,s=s^2-s-1);s \\ M. F. Hasler, Oct 06 2014

Formula

a(n) = a(n-1)^2-a(n-1)-1, a(0)=3.

a(n) ~ c^(2^n), where c = 2.07259396780115004655284076205241023281287049774423620992171834046728756... . - Vaclav Kotesovec, May 06 2015

Extensions

Edited by M. F. Hasler, Oct 06 2014