A076840 -id:A076840 - 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J. H. Conway and R. L. Graham, On Periodic Sequences Defined by Recurrences, unpublished, date?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

A076844 a(1) = a(2) = a(3) = 1; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

Original entry on oeis.org

1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 3, 5, 9, 5, 3, 1

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) + a(n-2) + 1)/a(n-3) (for n>3) has period 8. - James Propp, Nov 20 2002. This is the 8-cycle discovered by H. Todd - see Lyness, Note 1847. - 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.
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 1).

Cf. A076840, A076841, A076839, A076842, A076843.

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;

nxt[{a_,b_,c_}]:={b,c,(b+c+1)/a}; Transpose[NestList[nxt,{1,1,1},110]][[1]] (* or *) PadRight[{},110,{1,1,1,3,5,9,5,3}] (* Harvey P. Dale, Jan 13 2015 *)
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 1},{1, 1, 1, 3, 5, 9, 5, 3},105] (* Ray Chandler, Aug 25 2015 *)

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 *)

A076369 a(n) = n + mu(n), where mu is the Moebius function (A008683).

Original entry on oeis.org

2, 1, 2, 4, 4, 7, 6, 8, 9, 11, 10, 12, 12, 15, 16, 16, 16, 18, 18, 20, 22, 23, 22, 24, 25, 27, 27, 28, 28, 29, 30, 32, 34, 35, 36, 36, 36, 39, 40, 40, 40, 41, 42, 44, 45, 47, 46, 48, 49, 50, 52, 52, 52, 54, 56, 56, 58, 59, 58, 60, 60, 63, 63, 64, 66, 65, 66, 68, 70, 69, 70, 72

Offset: 1

Reinhard Zumkeller and Labos Elemer, Oct 14 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000027, A005117, A008683, A013929, A063015, A076840.

a[n_] := n + MoebiusMu[n]; Array[a, 100] (* Amiram Eldar, Apr 09 2025 *)

a(n) = n + moebius(n); \\ Amiram Eldar, Apr 09 2025

a(n) = n + mu(m) = A000027(n) + A008683(n).

a(n) = n iff n is not squarefree: a(A013929(k)) = A013929(k) and a(A005117(k)) <> A005117(k).

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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A076844 a(1) = a(2) = a(3) = 1; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

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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).

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A076369 a(n) = n + mu(n), where mu is the Moebius function (A008683).

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