A064098 - OEIS

%I #87 Nov 08 2024 07:20:56

%S 1,1,1,2,5,29,433,37666,48928105,5528778008357,811537892743746482789,

%T 13460438563050022083842073547074914,

%U 32770967840592833551621556305285371426044732591005957081

%N a(n+1) = (a(n)^2 + a(n-1)^2)/a(n-2), with a(1) = a(2) = a(3) = 1.

%C This sequence was introduced by Dana Scott but possibly studied earlier by others. - _James Propp_, Jan 27 2005

%C Sequence gives the upper-left entries of the respective matrices

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

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

%C 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). - _James Propp_, Jan 27 2005

%C 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_, Jul 28 2002

%C 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

%C A subsequence of the Markoff numbers A002559. - _Andrew Hone_, Jan 16 2006

%C The recursion exhibits the Laurent phenomenon. Let F(n) = Fibonacci(n), e(n) = F(n) - 1, a(1) = a1, a(2) = a2, a(3) = a3, a(n) = (a(n-1)^2 + a(n-3)^2) / a(n-3), b(n) = a(n) * a1^e(n-1) * a2^e(n-2) * a3^e(n-3). Then b(n) for n > 1 is an irreducible polynomial in {a1^2, a2^2, a3^2}, b(n) = (b(n-1)^2 + (b(n-2) * a1^F(n-4) * a2^F(n-5) * a3^F(n-6))^2) / b(n-3), and a(n) = a(n-1) * a(n-2) * (a1^2 + a2^2 + a3^2) / (a1 * a2 * a3) - a(n-3). - _Michael Somos_, Jan 12 2013

%C Starting with n = 5, a(n) is the largest number in row n - 5 of the Markov tree, A368546. These numbers are obtained by descending the tree in alternating right and left steps; their Farey indices (see A368546 for the definition) are ratios of successive Fibonacci numbers, 1/2, 2/3, 3/5, 5/8, ... See Aigner, Proposition 10.2. - _Wouter Meeussen_ and _William P. Orrick_, Feb 11 2024

%D Martin Aigner, Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings. Springer, 2013. x+257 pp. ISBN: 978-3-319-00887-5; 978-3-319-00888-2 MR3098784

%H Harry J. Smith, <a href="/A064098/b064098.txt">Table of n, a(n) for n = 1..18</a>

%H Martin Aigner, <a href="https://archive.org/details/markovstheorem100000aign">Markov's theorem and 100 years of the uniqueness conjecture. A mathematical journey from irrational numbers to perfect matchings</a>, [archive.org copy of the book].

%H S. Fomin and A. Zelevinsky, <a href="http://dx.doi.org/10.1006/aama.2001.0770">The Laurent Phenomenon</a>, Advances in Applied Mathematics, 28 (2002), 119-144.

%H Andrew N. W. Hone, <a href="https://arxiv.org/abs/math/0601324">Diophantine non-integrability of a third order recurrence with the Laurent property</a>, arXiv:math/0601324 [math.NT], 2006.

%H Andrew N. W. Hone, <a href="https://doi.org/10.1088/0305-4470/39/12/L01">Diophantine non-integrability of a third order recurrence with the Laurent property</a>, J. Phys. A: Math. Gen. 39 (2006), L171-L177.

%H Andrew N. W. Hone, <a href="https://arxiv.org/abs/2109.08217">Growth of Mahler measure and algebraic entropy of dynamics with the Laurent property</a>, arXiv:2109.08217 [math.NT], 2021.

%H KöMaL-Mathematical and Physical Journal for Secondary Schools, <a href="https://www.komal.hu/verseny/2001-04/mat.e.shtml">New advanced problems: proposed problem A265</a>, April 2001.

%H L. J. Mordell, <a href="https://doi.org/10.1112/jlms/s1-28.4.500">On the integer solutions of the equation x^2+y^2+z^2+2xyz=n</a>, J. Lond. Math. Soc. 28 (1953), 500-510.

%H J. Propp, <a href="http://arxiv.org/abs/math/0511633">The combinatorics of frieze patterns and Markoff numbers</a>, arXiv:math/0511633 [math.CO], 2005-2008.

%H Matthew Christopher Russell, <a href="https://doi.org/10.7282/T3MC926D">Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

%F Conjecture: lim_{n -> infinity} log(log(a(n)))/n exists = 0.48.... - _Benoit Cloitre_, Aug 07 2002. This is true - see below.

%F 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_, Jan 16 2006

%F a(n) = 3 * a(n-1) * a(n-2) - a(n-3). a(4-n) = a(n) for all n in Z. - _Michael Somos_, Jan 12 2013

%F a(n) ~ 1/3 * c^(((1 + sqrt(5))/2)^n), where c = 1.2807717799265504005186306582930649245... . - _Vaclav Kotesovec_, May 06 2015

%e G.f. = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 29*x^6 + 433*x^7 + 37666*x^8 + ...

%p f:=proc(n) option remember; global K; local i;

%p if n <= K then 1

%p else add(f(n-i)^2,i=1..K-1)/f(n-K); fi; end;

%p K:=3;

%p [seq(f(n),n=1..10)]; # _N. J. A. Sloane_, Mar 17 2017

%t a[n_]:= (a[n-1]^2 +a[n-2]^2)/a[n-3]; a[1]=a[2]=a[3]=1; Array[a, 13] (* Or *)

%t a[n_]:= 3*a[n-1]*a[n-2] - a[n-3]; a[1]= a[2]= a[3]= 1; Array[a, 13] (* _Robert G. Wilson v_, Dec 26 2012 *)

%o (PARI) {a(n) = if( n<1, n = 4-n); if( n<4, 1, 3 * a(n-1) * a(n-2) - a(n-3))}; /* _Michael Somos_, Jan 12 2013 */

%o (PARI) { a=a3=a2=a1=1; for (n = 1, 18, if (n>3, a=(a1^2 + a2^2)/a3; a3=a2; a2=a1; a1=a); write("b064098.txt", n, " ", a) ) } /* _Harry J. Smith_, Sep 06 2009 */

%o (Magma) [n le 3 select 1 else (Self(n-1)^2 + Self(n-2)^2)/Self(n-3): n in [1..16]]; // _G. C. Greubel_, Nov 07 2024

%o (SageMath)

%o def A064098(n):

%o     def a(n): return 1 if n<4 else (a(n-1)^2 + a(n-2)^2)/a(n-3)

%o     return a(n)

%o [A064098(n) for n in range(16)] # _G. C. Greubel_, Nov 07 2024

%Y Cf. A002559, A072878, A072879, A072880.

%Y Markov tree: A327345, A368546.

%K nonn

%O 1,4

%A _Santi Spadaro_, Sep 16 2001

%E Entry improved by comments from _Michael Somos_, Sep 25 2001