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

1,3

Comments

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

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

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J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?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

Table of n, a(n) for n=1..105.

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).

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

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;

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 *)

Crossrefs

Cf. A076840, A076841, A076844, A076824, A105736 - A105746.

See A335688/A335689 for a very similar nonperiodic sequence.

Sequence in context: A306239 A159455 A105734 * A092542 A321305 A339178

Adjacent sequences: A076836 A076837 A076838 * A076840 A076841 A076842

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 21 2002

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

Status

approved