A074066 -id:A074066 - cp's OEIS frontend

A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

0, 2, 1, 3, 5, 4, 6, 8, 7, 9, 11, 10, 12, 14, 13, 15, 17, 16, 18, 20, 19, 21, 23, 22, 24, 26, 25, 27, 29, 28, 30, 32, 31, 33, 35, 34, 36, 38, 37, 39, 41, 40, 42, 44, 43, 45, 47, 46, 48, 50, 49, 51, 53, 52, 54, 56, 55, 57, 59, 58, 60, 62, 61, 63, 65, 64, 66, 68, 67, 69, 71, 70

Offset: 0

Reinhard Zumkeller, Oct 15 2001

a(a(n)) = n (a self-inverse permutation).
Take natural numbers, exchange trisections starting with 1 and 2.
Lodumo_3 of A080425. - Philippe Deléham, Apr 26 2009
From Franck Maminirina Ramaharo, Jul 27 2018: (Start)
The sequence is A008585 interleaved with A016789 and A016777.
a(n) is also obtained as follows: write n in base 3; if the rightmost digit is '1', then replace it with '2' and vice versa; convert back to decimal. For example a(14) = a('11'2') = '11'1' = 13 and a(10) = a('10'1') = '10'2' = 11. (End)
A permutation of the nonnegative integers partitioned into triples [3*k-3, 3*k-1, 3*k-2] for k > 0. - Guenther Schrack, Feb 05 2020

			From _Franck Maminirina Ramaharo_, Jul 27 2018: (Start)
Interleave 3 sequences:
A008585: 0.....3.....6.....9.......12.......15........
A016789: ..2.....5.....8.....11.......14.......17.....
A016777: ....1.....4.....7......10.......13.......16..
(End)

Muniru A Asiru, Table of n, a(n) for n = 0..3000
Eric Weisstein's World of Mathematics, Alternating Permutations.
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. A001477, A004442, A074066, A080425, A080782, A102283, A143097.

a:=[0,2,1,3];; for n in [5..100] do a[n]:=a[n-1]+a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Jul 27 2018

[2*n - 3 - 3*((n-2) div 3): n in [0..80]]; // Vincenzo Librandi, Aug 05 2018

A064429:=n->2*n-3-3*floor((n-2)/3): seq(A064429(n), n=0..100); # Wesley Ivan Hurt, Nov 30 2013

Table[2 n - 3 - 3 Floor[(n - 2)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 30 2013 *)
{#+1,#-1,#}[[Mod[#,3,1]]]&/@Range[0, 100] (* Federico Provvedi, May 11 2021 *)
LinearRecurrence[{1,0,1,-1},{0,2,1,3},80] (* or *) {#[[1]],#[[3]],#[[2]]}&/@Partition[Range[0,80],3]//Flatten (* Harvey P. Dale, Mar 28 2025 *)

a(n) = 2*n-3-3*((n-2)\3); \\ Altug Alkan, Oct 06 2017

a(n) = A080782(n+1) - 1.
a(n) = n - 2*sin(4*Pi*n/3)/sqrt(3). - Jaume Oliver Lafont, Dec 05 2008
a(n) = A001477(n) + A102283(n). - Jaume Oliver Lafont, Dec 05 2008
a(n) = lod_3(A080425(n)). - Philippe Deléham, Apr 26 2009
G.f.: x*(2 - x + 2*x^2)/((1 + x + x^2)*(1 - x)^2 ). - R. J. Mathar, Feb 20 2011
a(n) = 2*n - 3 - 3*floor((n-2)/3). - Wesley Ivan Hurt, Nov 30 2013
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3. - Wesley Ivan Hurt, Oct 06 2017
E.g.f.: x*exp(x) + (2*sin((sqrt(3)*x)/2))/(exp(x/2)*sqrt(3)). - Franck Maminirina Ramaharo, Jul 27 2018
From Guenther Schrack, Feb 05 2020: (Start)
a(n) = a(n-3) + 3 with a(0)=0, a(1)=2, a(2)=1 for n > 2;
a(n) = n + (w^(2*n) - w^n)*(1 + 2*w)/3 where w = (-1 + sqrt(-3))/2. (End)
Sum_{n>=1} (-1)^n/a(n) = log(2)/3. - Amiram Eldar, Jan 31 2023

A092486 Take natural numbers, exchange first and third quadrisection.

Original entry on oeis.org

3, 2, 1, 4, 7, 6, 5, 8, 11, 10, 9, 12, 15, 14, 13, 16, 19, 18, 17, 20, 23, 22, 21, 24, 27, 26, 25, 28, 31, 30, 29, 32, 35, 34, 33, 36, 39, 38, 37, 40, 43, 42, 41, 44, 47, 46, 45, 48, 51, 50, 49, 52, 55, 54, 53, 56, 59, 58, 57, 60, 63, 62, 61, 64, 67, 66, 65, 68, 71, 70, 69, 72

Offset: 0

Ralf Stephan, Apr 04 2004

Harry J. Smith, Table of n, a(n) for n = 0..20003
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A080412, A064429, A074066, A080782, A159966.

Flatten[Partition[Range[80],4]/.{a_,b_,c_,d_}->{c,b,a,d}] (* Harvey P. Dale, Aug 12 2012 *)

{ f="b092486.txt"; for (n=0, 5000, a0=4*n + 3; a1=a0 - 1; a2=a1 - 1; a3=a0 + 1; write(f, 4*n, " ", a0); write(f, 4*n+1, " ", a1); write(f, 4*n+2, " ", a2); write(f, 4*n+3, " ", a3); ); } \\ Harry J. Smith, Jun 21 2009

G.f.: (3-4*x+3*x^2)/((1+x^2)*(1-x)^2).
a(4n) = 4n+3, a(4n+1) = 4n+2, a(4n+2) = 4n+1, a(4n+3) = 4n+4.
a(n) = n+1+i^n+(-i)^n, where i is the imaginary unit. - Bruno Berselli, Feb 08 2011
From Wesley Ivan Hurt, May 09 2021: (Start)
a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-a(n-4).
a(n) = 1 + n + 2*cos(n*Pi/2). (End)
Sum_{n>=0} (-1)^n/a(n) = log(2) (A002162). - Amiram Eldar, Nov 28 2023

A330396 Permutation of the nonnegative integers partitioned into triples [3k+2, 3k+1, 3*k] for k >= 0.

Original entry on oeis.org

2, 1, 0, 5, 4, 3, 8, 7, 6, 11, 10, 9, 14, 13, 12, 17, 16, 15, 20, 19, 18, 23, 22, 21, 26, 25, 24, 29, 28, 27, 32, 31, 30, 35, 34, 33, 38, 37, 36, 41, 40, 39, 44, 43, 42, 47, 46, 45, 50, 49, 48, 53, 52, 51, 56, 55, 54, 59, 58, 57, 62, 61, 60, 65, 64, 63, 68, 67, 66, 71, 70, 69, 74, 73, 72, 77, 76, 75, 80, 79, 78, 83, 82

Offset: 0

Guenther Schrack, Mar 03 2020

nonn

Partition the nonnegative integer sequence into triples starting with (0,1,2); transpose the first and third elements of the triple, repeat for all triples.
A self-inverse sequence: a(a(n)) = n.
The sequence is an interleaving of A016789 with A016777 and with A008585, in that order.

Guenther Schrack, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the nonnegative integers
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A001477, A064429, A236348, A004482, A004483.
Fixed point sequence: A016777.
Relationships:
  a(n) = a(n-1) - 1 + 6*A079978(n).
  a(n) = 2*a(n-1) - a(n-2) + 6*A049347(n).
  a(n) = A074066(n+2) - 2.
  a(n) = A113655(n+1) - 1.

a = zeros(1,10000);
w = (-1+sqrt(-3))/2;
fprintf('0 2\n');
for n = 1:10000
   a(n) = int64((3*n + 2*w^(2*n)*(w + 2) + 2*w^n*(1 - w))/3);
   fprintf('%i %i\n',n,a(n));
end

G.f.: (2 - x - x^2 + 3*x^3)/((x-1)^2*(1 + x + x^2)). [corrected by Georg Fischer, Apr 17 2020]
Linear recurrence: a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
Simple recursion: a(n) = a(n-3) + 3 for n > 2 with a(0) = 2, a(1) = 1, a(2) = 0.
Negative domain: a(-n) = -(a(n-1) + 1).
Explicit formulas:
  a(n) = n + 2 - 2*(n mod 3).
  a(n) = 2 - n + 6*floor(n/3).
  a(n) = n + 2*(w^(2*n)*(2 + w) + w^n*(1 - w))/3 where w = (-1 + sqrt(-3))/2.

A074067 Zigzag modulo 5.

Original entry on oeis.org

1, 2, 7, 6, 5, 4, 3, 12, 11, 10, 9, 8, 17, 16, 15, 14, 13, 22, 21, 20, 19, 18, 27, 26, 25, 24, 23, 32, 31, 30, 29, 28, 37, 36, 35, 34, 33, 42, 41, 40, 39, 38, 47, 46, 45, 44, 43, 52, 51, 50, 49, 48, 57, 56, 55, 54, 53, 62, 61, 60, 59, 58, 67, 66, 65, 64, 63, 72, 71

Offset: 1

Reinhard Zumkeller, Aug 17 2002

Eric Weisstein's World of Mathematics, Alternating Permutations.
Reinhard Zumkeller, Illustration for A074066-A074068.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A074066, A074068.

a074067 n = a074067_list !! (n-1)
a074067_list = 1 : 2 : xs where xs = 7 : 6 : 5 : 4 : 3 : map (+ 5) xs
-- Reinhard Zumkeller, Feb 21 2011

{1, 2}~Join~Flatten[Reverse /@ Partition[Range[3, 72], 5]] (* after Harvey P. Dale at A074066, or *)
{1, 2}~Join~Table[5 Floor[n/5] + 10 Floor[#/3] - # &@ Mod[n, 5], {n, 3, 69}] (* Michael De Vlieger, May 25 2016 *)
LinearRecurrence[{1,0,0,0,1,-1},{1,2,7,6,5,4,3,12},70] (* Harvey P. Dale, Jun 18 2025 *)

a(a(n)) = n (a self-inverse permutation).
For n > 1: a(n) = n iff n == 0 modulo 5.
a(n) = 5*floor(n/5) + 10*floor((n mod 5)/3) - (n mod 5) for n > 2; a(n) = n for n <= 2.
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 8. - Chai Wah Wu, May 25 2016
G.f.: x+2*x + x^3*(7-x-x^2-x^3-x^4+2*x^5) / ( (x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, May 22 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, May 11 2025

A074068 Zigzag modulo 7.

Original entry on oeis.org

1, 2, 3, 10, 9, 8, 7, 6, 5, 4, 17, 16, 15, 14, 13, 12, 11, 24, 23, 22, 21, 20, 19, 18, 31, 30, 29, 28, 27, 26, 25, 38, 37, 36, 35, 34, 33, 32, 45, 44, 43, 42, 41, 40, 39, 52, 51, 50, 49, 48, 47, 46, 59, 58, 57, 56, 55, 54, 53, 66, 65, 64, 63, 62, 61, 60, 73, 72, 71

Offset: 1

Reinhard Zumkeller, Aug 17 2002

Eric Weisstein's World of Mathematics, Alternating Permutations.
Reinhard Zumkeller, Illustration for A074066-A074068.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).
Index entries for sequences that are permutations of the natural numbers.

Cf. A002162, A074066, A074067.

a074068 n = a074068_list !! (n-1)
a074068_list = 1 : 2 : 3 : xs where
   xs = 10 : 9 : 8 : 7 : 6 : 5 : 4 : map (+ 7) xs
-- Reinhard Zumkeller, Feb 21 2011

Range[3]~Join~Flatten[Reverse /@ Partition[Range[4, 73], 7]] (* after Harvey P. Dale at A074066, or *)
Range[3]~Join~Table[7 Floor[n/7] + 14 Floor[#/4] - # &@ Mod[n, 7], {n, 4, 69}] (* Michael De Vlieger, May 25 2016 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{1,2,3,10,9,8,7,6,5,4,17},80] (* Harvey P. Dale, Jan 19 2025 *)

a(a(n)) = n (a self-inverse permutation).
For n > 1: a(n) = n iff n == 0 modulo 7.
a(n) = 7*floor(n/7) + 14*floor((n mod 7)/4) - (n mod 7) for n > 3; a(n) = n for n <= 3.
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 11. - Chai Wah Wu, May 25 2016
G.f.: x+2*x^2+3*x^3 + x^4*(10-x-x^2-x^3-x^4-x^5-x^6+3*x^7) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, May 22 2019
Sum_{n>=1} (-1)^(n+1)/a(n) = log(2) (A002162). - Amiram Eldar, May 11 2025

cp's OEIS Frontend

A064429 a(n) = floor(n / 3) * 3 + sign(n mod 3) * (3 - n mod 3).

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A092486 Take natural numbers, exchange first and third quadrisection.

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A330396 Permutation of the nonnegative integers partitioned into triples [3*k+2, 3*k+1, 3*k] for k >= 0.

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A074067 Zigzag modulo 5.

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A074068 Zigzag modulo 7.

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A330396 Permutation of the nonnegative integers partitioned into triples [3k+2, 3k+1, 3*k] for k >= 0.