A135352 -id:A135352 - cp's OEIS frontend

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

N. J. A. Sloane, Nov 21 2002

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

J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?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

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).

Cf. A076840, A076841, A076844, A076824, A105736 - A105746.
See A335688/A335689 for a very similar nonperiodic sequence.
This sequence and A135352 are bisections of each other.

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;

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 *)

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

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

A139434 Frieze pattern with 4 rows, read by diagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2

Offset: 0

N. J. A. Sloane, Jun 09 2008

Period 20: repeat [1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 3, 2, 1]. - Wesley Ivan Hurt, Jun 05 2016

Every frieze is defined by its quiddity (the row below the row of 1's), which corresponds to the counts of triangles at vertices of a dissection of a regular polygon. The quiddity of this frieze is A135352. One can say that this frieze pattern has width 2 (not counting the rows of 1's), 4, or 5 (implying the additional row of 0's; this is also the period of the pattern and the number of vertices in the dissected polygon), depending on the convention. In any case, friezes of given width are enumerated by A000207 if we identify shifts and mirror images, otherwise by A000108. A000207(3) = 1 means that this is the only frieze of this width, and it has A000108(3) = 5 different horizontal shifts or reflections. The A000207(4) = 3 friezes having width 1 greater than this one are A139438, A139458, and one more with quiddity 1, 3, 1, 3, 1, 3, ... (currently not in the OEIS). The only frieze having width 1 less than this one has quiddity 1, 2, 1, 2, ... (A245477 can be interpreted as representing that frieze pattern). - Andrey Zabolotskiy, Jan 30 2024

			The frieze pattern is
...1 1 1 1 1 1 1 ...
....1 2 2 1 3 1 2 ...
.....1 3 1 2 2 1 3 ...
......1 1 1 1 1 1 1 ...

J. H. Conway and R. K. Guy, The Book of Numbers. New York: Springer-Verlag, p. 97, 1996.

Cf. A139438, A139458, A135352, A000108, A000207, A245477.

cp's OEIS Frontend

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).

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