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).
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, 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, 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
Offset: 1
References
- J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?
- Martin Gardner, The Magic Numbers of Dr Matrix, Prometheus Books, 1985, pages 198 and 305.
Links
- Sergey Fomin and Andrei Zelevinsky, Cluster algebras II: Finite type classification, arXiv:math/0208229 [math.RA], 2002-2003.
- Jonny Griffiths, Lyness cycles, elliptic curves, and Hikorsky triangles, MSc Thesis, University of East Anglia, Norwich, UK, Department of Mathematics, Feb 2012.
- V. L. Kocic, G. Ladas, and I. W. Rodrigues, On Rational Recursive Sequences, J. Math. Anal. Appl., 173 (1993), 127-157.
- R. C. Lyness, Note 1581. Cycles, Math. Gazette, 26 (1942), 62.
- R. C. Lyness, Note 1847. Cycles, Math. Gaz., 29 (1945), 231-233.
- R. C. Lyness, Note 2952. Cycles, Math. Gaz., 45 (1961), 207-209.
- S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, arXiv:1008.3359 [math.AG], 2010-2011.
- S. Morier-Genoud, V. Ovsienko and S. Tabachnikov, 2-frieze patterns and the cluster structure of the space of polygons, Annales de l'institut Fourier, 62 no. 3 (2012), 937-987.
- Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).
Crossrefs
Programs
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Maple
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;
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Mathematica
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 *)
Formula
Periodic with period 5.
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
Extensions
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
Comments