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A076839 A simple example of the Lyness 5-cycle: a(1) = a(2) = 1; a(n) = (a(n-1)+1)/a(n-2) (for n>2).

%I A076839 #92 May 15 2025 05:28:56
%S A076839 1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,
%T A076839 2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,
%U A076839 3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2,1,1,2,3,2
%N A076839 A simple example of the Lyness 5-cycle: a(1) = a(2) = 1; a(n) = (a(n-1)+1)/a(n-2) (for n>2).
%C A076839 Any sequence a(1),a(2),a(3),... defined by the recurrence a(n) = (a(n-1)+1)/a(n-2) (for n>2) has period 5. The theory of cluster algebras currently being developed by Fomin and Zelevinsky gives a context for these facts, but it doesn't really explain them in an elementary way. - _James Propp_, Nov 20 2002
%C A076839 Equivalently, for n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, with a(1)=1, a(2)=1. - _Ray Chandler_, May 16 2024
%D A076839 J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?
%D A076839 Martin Gardner, The Magic Numbers of Dr Matrix, Prometheus Books, 1985, pages 198 and 305.
%H A076839 Sergey Fomin and Andrei Zelevinsky, <a href="https://arxiv.org/abs/math/0208229">Cluster algebras II: Finite type classification</a>, arXiv:math/0208229 [math.RA], 2002-2003.
%H A076839 Jonny Griffiths, <a href="http://www.s253053503.websitehome.co.uk/jg-msc-uea/thesis-final-11-2-2012.pdf">Lyness cycles, elliptic curves, and Hikorsky triangles</a>, MSc Thesis, University of East Anglia, Norwich, UK, Department of Mathematics, Feb 2012.
%H A076839 V. L. Kocic, G. Ladas, and I. W. Rodrigues, <a href="https://doi.org/10.1006/jmaa.1993.1057">On Rational Recursive Sequences</a>, J. Math. Anal. Appl., 173 (1993), 127-157.
%H A076839 R. C. Lyness, <a href="https://www.jstor.org/stable/3606036">Note 1581. Cycles</a>, Math. Gazette, 26 (1942), 62.
%H A076839 R. C. Lyness, <a href="https://www.jstor.org/stable/3609268">Note 1847. Cycles</a>, Math. Gaz., 29 (1945), 231-233.
%H A076839 R. C. Lyness, <a href="https://www.jstor.org/stable/3612778">Note 2952. Cycles</a>, Math. Gaz., 45 (1961), 207-209.
%H A076839 S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, <a href="http://arxiv.org/abs/1008.3359">2-frieze patterns and the cluster structure of the space of polygons</a>, arXiv:1008.3359 [math.AG], 2010-2011.
%H A076839 S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, <a href="https://doi.org/10.5802/aif.2713">2-frieze patterns and the cluster structure of the space of polygons</a>, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987.
%H A076839 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,1).
%F A076839 Periodic with period 5.
%F A076839 a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=2, a(n)=a(n-5). - _Harvey P. Dale_, Jan 17 2013
%p A076839 a := 1; b := 1; f := proc(n) option remember; global a,b; if n=1 then a elif n=2 then b else (f(n-1)+1)/f(n-2); fi; end;
%t A076839 RecurrenceTable[{a[1]==a[2]==1,a[n]==(a[n-1]+1)/a[n-2]},a,{n,110}] (* or *) LinearRecurrence[{0,0,0,0,1},{1,1,2,3,2},110] (* _Harvey P. Dale_, Jan 17 2013 *)
%Y A076839 Cf. A076840, A076841, A076844, A076824, A105736 - A105746.
%Y A076839 See A335688/A335689 for a very similar nonperiodic sequence.
%Y A076839 This sequence and A135352 are bisections of each other.
%K A076839 nonn,easy
%O A076839 1,3
%A A076839 _N. J. A. Sloane_, Nov 21 2002
%E A076839 Thanks to _Michael Somos_ for pointing out the Kocic et al. (1993) reference. Also I deleted some useless comments. - _N. J. A. Sloane_, Jul 19 2020