A079354 -id:A079354 - cp's OEIS frontend

A080782 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

1, 3, 2, 4, 6, 5, 7, 9, 8, 10, 12, 11, 13, 15, 14, 16, 18, 17, 19, 21, 20, 22, 24, 23, 25, 27, 26, 28, 30, 29, 31, 33, 32, 34, 36, 35, 37, 39, 38, 40, 42, 41, 43, 45, 44, 46, 48, 47, 49, 51, 50, 52, 54, 53, 55, 57, 56, 58, 60, 59, 61, 63, 62, 64, 66, 65, 67, 69, 68

Offset: 1

Benoit Cloitre, Mar 07 2003

Permutation of the integers: exchange trisections starting with 2 and 3.

a(a(n)) = n. - Reinhard Zumkeller, Oct 29 2004

Guenther Schrack, Table of n, a(n) for n = 1..10000
Robbert Fokkink and Gandhar Joshi, On Cloitre's hiccup sequences, arXiv:2507.16956 [math.CO], 2025. See p. 2.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A064429, A079357, A079354, A080783, A064437.

Cf. A004442, A080412.

Array[#+Mod[#+1,3]&,70,0] (* or *) LinearRecurrence[{1,0,1,-1},{1,3,2,4},70] (* Harvey P. Dale, Mar 29 2013 *)
{#,#+1,#-1}[[Mod[#,3,1]]]&/@Range[99] (* Federico Provvedi, May 15 2021 *)

a(n) = A064429(n-1) + 1.
a(n) - n is periodic with period 3.
G.f.: x*(1+2*x-x^2+x^3)/(1-x-x^3+x^4). - Jaume Oliver Lafont, Mar 24 2009
a(0)=1, a(1)=3, a(2)=2, a(3)=4, a(n)=a(n-1)+0*a(n-2)+a(n-3)-a(n-4). - Harvey P. Dale, Mar 29 2013
a(n) = n + (2/sqrt(3))*sin(2*(n+2)*Pi/3). - Wesley Ivan Hurt, Sep 26 2017
From Guenther Schrack, Oct 23 2019: (Start)
a(n) = a(n-3) + 3 with a(1) = 1, a(2) = 3, a(3) = 2 for n > 3.
a(n) = n - (w^(2*n)*(2 + w) + w^n*(1 - w))/3 where w = (-1 + sqrt(-3))/2. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) - log(2)/3. - Amiram Eldar, Jan 31 2023
From Charles L. Hohn, Sep 03 2024: (Start)
a(n) = n-1+n%3.
a(n) = A375336(n-2, 1) for n >= 6. (End)

A079357 a(1)=1; a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

Original entry on oeis.org

1, 5, 9, 13, 12, 16, 20, 24, 23, 27, 31, 30, 29, 33, 37, 36, 40, 44, 48, 47, 51, 55, 54, 53, 57, 61, 60, 64, 63, 62, 61, 65, 64, 68, 72, 71, 70, 74, 78, 77, 81, 85, 89, 88, 92, 96, 95, 94, 98, 102, 101, 105, 104, 103, 102, 106, 105, 109, 113, 112, 111, 110, 109, 108, 107

Offset: 1

Benoit Cloitre, Feb 14 2003

If, in the defining recurrence, the rule a(n)=a(n-1)+4 when n is not already in the sequence is generalized to a(n)=a(n-1)+k, then the resulting sequence ultimately becomes periodic with period 1,3,10,35 for k=1,2,3,4, respectively. - John W. Layman, Apr 15 2003

Cf. A080782, A079354, A079355, A080783, A064437.

It appears that, for n >= 219, a(n)=n+b(n) where b(n) is the period-35 sequence (-1, 2, 5, 3, 6, 9, 7, 5, 8, 11, 9, 12, 10, 8, 6, 9, 7, 10, 13, 11, 9, 7, 5, 3, 1, -1, -3, -5, -7, -9, -11, -13, -10, -7, -4).

More terms from John W. Layman, Apr 15 2003

A080783 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

Original entry on oeis.org

1, 6, 11, 16, 21, 20, 25, 30, 35, 40, 39, 44, 49, 54, 59, 58, 63, 68, 73, 72, 71, 76, 81, 86, 85, 90, 95, 100, 105, 104, 109, 114, 119, 124, 123, 128, 133, 138, 137, 136, 141, 146, 151, 150, 155, 160, 165, 170, 169, 174, 179, 184, 189, 188, 193, 198

Offset: 1

Benoit Cloitre, Mar 07 2003

nonn

Cf. A079357, A079354, A080782, A064437.

lst={1};i=2;Do[If[MemberQ[lst,i],AppendTo[lst,Last[lst]-1], AppendTo[ lst,Last[lst]+5]];i++,{60}];lst (* Harvey P. Dale, Aug 20 2011 *)

Conjectured to be asymptotic to 3n as n -> infinity.

cp's OEIS Frontend

A080782 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+2 otherwise.

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A079357 a(1)=1; a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+4 otherwise.

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A080783 a(1)=1, a(n)=a(n-1)-1 if n is already in the sequence, a(n)=a(n-1)+5 otherwise.

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