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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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%I #61 Sep 27 2024 00:02:13

%S 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,

%T 26,28,30,29,31,33,32,34,36,35,37,39,38,40,42,41,43,45,44,46,48,47,49,

%U 51,50,52,54,53,55,57,56,58,60,59,61,63,62,64,66,65,67,69,68

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

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

%C a(a(n)) = n. - _Reinhard Zumkeller_, Oct 29 2004

%H Guenther Schrack, <a href="/A080782/b080782.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>.

%F a(n) = A064429(n-1) + 1.

%F a(n) - n is periodic with period 3.

%F G.f.: x*(1+2*x-x^2+x^3)/(1-x-x^3+x^4). - _Jaume Oliver Lafont_, Mar 24 2009

%F 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

%F a(n) = n + (2/sqrt(3))*sin(2*(n+2)*Pi/3). - _Wesley Ivan Hurt_, Sep 26 2017

%F From _Guenther Schrack_, Oct 23 2019: (Start)

%F a(n) = a(n-3) + 3 with a(1) = 1, a(2) = 3, a(3) = 2 for n > 3.

%F a(n) = n - (w^(2*n)*(2 + w) + w^n*(1 - w))/3 where w = (-1 + sqrt(-3))/2. (End)

%F Sum_{n>=1} (-1)^(n+1)/a(n) = 2*Pi/(3*sqrt(3)) - log(2)/3. - _Amiram Eldar_, Jan 31 2023

%F From _Charles L. Hohn_, Sep 03 2024: (Start)

%F a(n) = n-1+n%3.

%F a(n) = A375336(n-2, 1) for n >= 6. (End)

%t Array[#+Mod[#+1,3]&,70,0] (* or *) LinearRecurrence[{1,0,1,-1},{1,3,2,4},70] (* _Harvey P. Dale_, Mar 29 2013 *)

%t {#,#+1,#-1}[[Mod[#,3,1]]]&/@Range[99] (* _Federico Provvedi_, May 15 2021 *)

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

%Y Cf. A004442, A080412.

%K nonn,easy

%O 1,2

%A _Benoit Cloitre_, Mar 07 2003