A076844 -id:A076844 - 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J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?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

A076840 a(1) = a(2) = 1; a(n) = (a(n-1) + 1)/a(n-2) (for n>2, n odd), (a(n-1)^2 + 1)/a(n-2) (for n>2, n even).

Original entry on oeis.org

1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 2, 5, 3, 2, 1, 1, 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, n odd), (a(n-1)^2+1)/a(n-2) (for n>2, n even) has period 6. 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

Sergey Fomin and Andrei Zelevinsky, Cluster algebras II: Finite type classification, arXiv:math/0208229 [math.RA], 2002-2003.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A076839, A076841, A076844.

a := 1; b := 1; f := proc(n) option remember; global a,b; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n mod 2 = 1 then RETURN((f(n-1)+1)/f(n-2)); fi; RETURN((f(n-1)^2+1)/f(n-2)); end;

LinearRecurrence[{0, 0, 0, 0, 0, 1}, {1, 1, 2, 5, 3, 2}, 105] (* Jean-François Alcover, Nov 22 2017 *)

A076841 a(1) = a(2) = 1; a(n) = (a(n-1)+1)/a(n-2) (for n>2, n odd), (a(n-1)^3+1)/a(n-2) (for n>2, n even).

Original entry on oeis.org

1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2, 9, 5, 14, 3, 2, 1, 1, 2

Offset: 1

N. J. A. Sloane, Nov 21 2002

nonn

Any sequence a(1),a(2),a(3),... defined by the recurrence a(n) = (a(n-1)+1)/a(n-2) (for n>2, n odd), (a(n-1)^3+1)/a(n-2) (for n>2, n even) has period 8. 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

Sergey Fomin and Andrei Zelevinsky, Cluster algebras II: Finite type classification
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A076839, A076840, A076844.

a := 1; b := 1; f := proc(n) option remember; global a,b; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n mod 2 = 1 then RETURN((f(n-1)+1)/f(n-2)); fi; RETURN((f(n-1)^3+1)/f(n-2)); end;

LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 2, 9, 5, 14, 3, 2},99] (* Ray Chandler, Aug 25 2015 *)

A076842 Numerators of sequence of fractions defined by a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1)+a(n-2)+1)/a(n-2).

Original entry on oeis.org

1, 1, 3, 5, 3, 9, 29, 71, 751, 5095, 117707, 8786157, 1495204807, 869159527657, 10563805576326283, 3058448407404141530011, 21945047981309619478239602921, 78849616271246494204031567399858651677, 163732240861295155285004461265839986271194799721161

Offset: 1

N. J. A. Sloane, Nov 21 2002

			1, 1, 3, 5, 3, 9/5, 29/15, 71/27, 751/261, 5095/2059, 117707/53321, ...

Cf. A076843, A076844.

a := 1; b := 1; f := proc(n) option remember; global a,b; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-2)); end;

A076843 Denominators of sequence of fractions defined by a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1)+a(n-2)+1)/a(n-2).

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 15, 27, 261, 2059, 53321, 3826345, 599717165, 344730727333, 4379034727152233, 1299571503802595847199, 9181632264980134793594508931, 32308854341042109701936418652172579113, 67117797048843876734329361265492957492346044762131

Offset: 1

N. J. A. Sloane, Nov 21 2002

			1, 1, 3, 5, 3, 9/5, 29/15, 71/27, 751/261, 5095/2059, 117707/53321, ...

Cf. A076842, A076844.

a := 1; b := 1; f := proc(n) option remember; global a,b; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-2)); end;

Denominator[RecurrenceTable[{a[1]==a[2]==1,a[n]==(a[n-1]+a[n-2]+1)/ a[n-2]},a,{n,20}]] (* Harvey P. Dale, Jul 20 2014 *)

A335690 a(1) = 1, a(2) = a(3) = 2; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

Original entry on oeis.org

1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4

Offset: 1

N. J. A. Sloane, Jul 19 2020

nonn

This is another illustration of the 8-cycle discovered by H. Todd - see Lyness, Note 1847. Compare A076844. - N. J. A. Sloane, Jul 19 2020

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.

Cf. A076839, A076844.

a := 1; b := 1; c := 1; f := proc(n) option remember; global a,b,c; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n=3 then RETURN(c); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-3)); end;

RecurrenceTable[{a[1]==1,a[2]==a[3]==2,a[n]==(a[n-1]+a[n-2]+1)/a[n-3]},a,{n,90}] (* or *) PadRight[{},90,{1,2,2,5,4,5,2,2}] (* Harvey P. Dale, May 28 2021 *)

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).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A076840 a(1) = a(2) = 1; a(n) = (a(n-1) + 1)/a(n-2) (for n>2, n odd), (a(n-1)^2 + 1)/a(n-2) (for n>2, n even).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A076841 a(1) = a(2) = 1; a(n) = (a(n-1)+1)/a(n-2) (for n>2, n odd), (a(n-1)^3+1)/a(n-2) (for n>2, n even).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A076842 Numerators of sequence of fractions defined by a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1)+a(n-2)+1)/a(n-2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

A076843 Denominators of sequence of fractions defined by a(1) = a(2) = 1; for n > 2, a(n) = (a(n-1)+a(n-2)+1)/a(n-2).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

A335690 a(1) = 1, a(2) = a(3) = 2; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica