A105746 -id:A105746 - 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J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?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

A105736 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=3.

Original entry on oeis.org

1, 3, 4, 4, 3, 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

Offset: 1

Zak Seidov, Apr 19 2005

nonn

The sequence depends on seed terms a(1) and a(2); if a(1)=1, a(3)=a(2)+1. All(?) sequences end with cycle={1,2,3,2,1} (or {2,4,6,4,2}, which essentially the same cycle) of length=5. This particular sequence merges with A076839, starting with 6th term = 1.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105737 - A105746.

More terms from Ray Chandler, May 17 2024

A105737 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=4.

Original entry on oeis.org

1, 4, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

The sequence depends on seed terms a(1) and a(2); if a(1)=1, a(3)=a(2)+1. All(?) sequences end with cycle={1,2,3,2,1} (or {2,4,6,4,2}, which essentially the same cycle) of length=5. This particular sequence does not merge with A076839 or A105736 and ends with another cycle {2,4,6,4,2}.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105745 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1, a(2)=12.

Original entry on oeis.org

1, 12, 13, 4, 9, 9, 4, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4

Offset: 1

Zak Seidov, Apr 19 2005

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

A[1]:= 1: A[2]:= 12:
for n from 3 to 100 do
  R:= map(rhs@op, [msolve(y^2=A[n-1]^2, 4*A[n-2])]);
  ys:= map(t -> (floor((A[n-1]-t)/(4*A[n-2]))+1)*4*A[n-2]+t, R);
  A[n]:= (min(ys)^2-A[n-1]^2)/(4*A[n-2]);
od:
seq(A[i],i=1..100); # Robert Israel, Oct 02 2020

LinearRecurrence[{0,0,0,0,1},{1,12,13,4,9,9,4,5,6,8,8,6,2,4,6,4,2},100] (* or *) PadRight[{1,12,13,4,9,9,4,5,6,8,8,6},100,{4,2,2,4,6}] (* Harvey P. Dale, May 01 2025 *)

a(n)=a(n-5) for n >= 18. - Robert Israel, Oct 02 2020

G.f.: x*(4*x^16 + 4*x^15 + 2*x^14 + 2*x^13 + 3*x^12 - 2*x^11 + x^10 + x^9 - 2*x^8 + 8*x^7 + 8*x^6 - 8*x^5 - 9*x^4 - 4*x^3 - 13*x^2 - 12*x - 1)/(x^5 - 1). - Chai Wah Wu, May 07 2024

More terms from Robert Israel, Oct 02 2020

A105738 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=5.

Original entry on oeis.org

1, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

The sequence depends on seed terms a(1) and a(2); if a(1)=1, a(3)=a(2)+1. All(?) sequences end with cycle={1,2,3,2,1} (or {2,4,6,4,2}, which essentially the same cycle) of length = 5. This particular sequence merges with A105737, starting with 2nd term = 5.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1)

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105739 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=6.

Original entry on oeis.org

1, 6, 7, 3, 4, 4, 3, 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

Offset: 1

Zak Seidov, Apr 19 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105740 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=7.

Original entry on oeis.org

1, 7, 8, 12, 8, 4, 4, 3, 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

Zak Seidov, Apr 19 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105741 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=8.

Original entry on oeis.org

1, 8, 9, 14, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105742 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=9.

Original entry on oeis.org

1, 9, 10, 16, 8, 3, 5, 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

Offset: 1

Zak Seidov, Apr 19 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105743 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=10.

Original entry on oeis.org

1, 10, 11, 6, 5, 1, 4, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

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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A105736 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=3.

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A105737 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=4.

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A105745 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1, a(2)=12.

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Programs

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A105738 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=5.

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A105739 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=6.

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A105740 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=7.

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Extensions

A105741 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=8.

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Crossrefs

Extensions

A105742 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=9.

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Author

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Extensions

A105743 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=10.

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A105736 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=3.

A105737 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=4.

A105745 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1, a(2)=12.

A105738 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=5.

A105739 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=6.

A105740 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=7.

A105741 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=8.

A105742 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=9.

A105743 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=10.