1,4

This sequence was introduced by Dana Scott but possibly studied earlier by others. - James Propp, Jan 27 2005

James Propp, Jan 27 2005, also comments: "Sequence gives the upper-left entries of the respective matrices

[1 1] [1 0] [2 1] [5 2] [29 12] [433 179] [37666 15571]

[1 2] [0 1] [1 1] [2 1] [12 5], [179 74], [15571 6437], ...

satisfying the recurrence M(n) = M(n-1) M(n-3)^(-1) M(n-1) (note that the Fibonacci numbers satisfy the additive version of this recurrence).

Define b(1)=b(2)=b(3)=1; b(n)=(b(n-1)+b(n-2))^2/b(n-3); then a(n) = sqrt(b(n)). - Benoit Cloitre (benoit7848c(AT)orange.fr), Jul 28 2002

Any 3 successive terms of the sequence satisfy the Markov equation x^2 + y^2 + z^2 = 3 xyz. Therefore from the 3rd term on this is a subsequence of the Markov numbers, A002559. Also, we conjecture that the limit of Log[Log[a[n]]/n is Log[(Sqrt[5]+1)/2]. - Martin Giese (martin.giese(AT)oeaw.ac.at), Oct 13 2005

Subsequence of Markoff numbers. - Andrew Hone (anwh(AT)kent.ac.uk), Jan 16 2006

Author?, "New advanced problems: problem A265.", Komal, April 2001

S. Fomin and A. Zelevinsky, The Laurent phenomenon, Adv. Appl. Math. 28 (2002) 119-144.

L. J. Mordell, On the integer solutions of the equation x^2+y^2+z^2+2xyz=n, J. Lond. Math. Soc. 28 (1953) 500-510.

A. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property

Conjecture : lim n -> infinity Log(Log(a(n)))/n exists = 0.48..... - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 07 2002. This is true - see below.

For this subsequence of the Markoff numbers, we have 2^(F(n-1)-1) < a(n) < 3^(F(n-1)-1) for n>4, where F(n) are the Fibonacci numbers with F(0)=0, F(1)=1, F(n+1)=F(n)+F(n-1). Hence log(log(a(n)))/n tends to log((1+sqrt(5))/2) as previously conjectured. - Andrew Hone (anwh(AT)kent.ac.uk), Jan 16 2006

(PARI) a(n)=if(n<4, n>0, 3*a(n-1)*a(n-2)-a(n-3))

Cf. A002559.

Sequence in context: A108367 A103592 A098026 this_sequence A098717 A059784 A000283

Adjacent sequences: A064095 A064096 A064097 this_sequence A064099 A064100 A064101

nonn

Santi Spadaro (spados(AT)katamail.com), Sep 16 2001

Entry improved by comments from Michael Somos, Sep 25 2001