login
Recurrence sequence a(n)=a(n-1)^2-a(n-1)-1, a(0)=3.
6

%I #21 Mar 20 2022 04:36:16

%S 3,5,19,341,115939,13441735781,180680260792773944179,

%T 32645356640144805339284259388335434039861,

%U 1065719310162246533488642668727242229836148490441005113524301742665845135502859459

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

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

%C 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

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

%F a(n) ~ c^(2^n), where c = 2.07259396780115004655284076205241023281287049774423620992171834046728756... . - _Vaclav Kotesovec_, May 06 2015

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

%o (PARI) a(n,s=3)=for(i=1,n,s=s^2-s-1);s \\ _M. F. Hasler_, Oct 06 2014

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

%K nonn

%O 0,1

%A _Artur Jasinski_, Sep 20 2008

%E Edited by _M. F. Hasler_, Oct 06 2014