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A064098 a(n+1) = (a(n)^2+a(n-1)^2)/a(n-2), with a(1) = a(2) = a(3) = 1. 9
1, 1, 1, 2, 5, 29, 433, 37666, 48928105, 5528778008357, 811537892743746482789, 13460438563050022083842073547074914, 32770967840592833551621556305285371426044732591005957081 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

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). - James Propp, Jan 27 2005

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

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

A subsequence of the Markoff numbers A002559. - Andrew Hone, Jan 16 2006

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

REFERENCES

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

Matthew Christopher Russell, Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences, PhD Dissertation, Mathematics Department, Rutgers University, May 2016; https://pdfs.semanticscholar.org/fdeb/e20954dacb7ec7a24afe2cf491b951c5a28d.pdf. Also (better) http://www.math.rutgers.edu/~zeilberg/Theses/MatthewRussellThesis.pdf

LINKS

Harry J. Smith, Table of n, a(n) for n=1,...,18

S. Fomin and A. Zelevinsky, The Laurent Phenomenon, Advances in Applied Mathematics, 28 (2002), 119-144.

A. Hone, Diophantine non-integrability of a third order recurrence with the Laurent property, arXiv:math/0601324 [math.NT], 2006.

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.

J. Propp, The combinatorics of frieze patterns and Markoff numbers, arXiv:math/0511633 [math.CO], 2005-2008.

FORMULA

Conjecture : lim n -> infinity Log(Log(a(n)))/n exists = 0.48..... - Benoit Cloitre, 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, Jan 16 2006

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

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

EXAMPLE

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

MAPLE

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

if n <= K then 1

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

K:=3;

[seq(f(n), n=1..10)]; # From N. J. A. Sloane, Mar 17 2017

MATHEMATICA

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

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 *)

PROG

(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 */

(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 */

CROSSREFS

Cf. A002559, A072878, A072879, A072880.

Sequence in context: A209428 A098026 A179823 * A181078 A265773 A098717

Adjacent sequences:  A064095 A064096 A064097 * A064099 A064100 A064101

KEYWORD

nonn,changed

AUTHOR

Santi Spadaro, Sep 16 2001

EXTENSIONS

Entry improved by comments from Michael Somos, Sep 25 2001

STATUS

approved

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Last modified March 25 11:34 EDT 2017. Contains 284076 sequences.