A004442 -id:A004442 - 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

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

Original entry on oeis.org

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)

A163540 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.

Original entry on oeis.org

0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 2, 2, 3, 0, 3, 3, 2, 1, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 0, 1, 2, 1, 0, 1, 2, 2, 3, 2, 1, 1, 0, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 3, 2, 3, 0, 0, 0, 1, 2, 1, 1, 0

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

Taking every sixteenth term gives the same sequence: (and similarly for all higher powers of 16 as well): a(n) = a(16*n).

A. Karttunen, Table of n, a(n) for n = 1..65536

a(n) = A163540(A008598(n)) = A004442(A163541(n)). See also A163542.

HC = {L[n_ /; IntegerQ[n/2]] :> {F[n], L[n], L[n + 1], R[n + 2]},
   R[n_ /; IntegerQ[(n + 1)/2]] :> {F[n], R[n], R[n + 3], L[n + 2]},
   R[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], F[n + 3]},
   L[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], F[n + 1]},
   F[n_ /; IntegerQ[n/2]] :> {L[n], R[n + 1], R[n], L[n + 3]},
   F[n_ /; IntegerQ[(n + 1)/2]] :> {R[n], L[n + 3], L[n], R[n + 1]}};
a[1] = F[0]; Map[(a[n_ /; IntegerQ[(n - #)/16]] :=
    Part[Flatten[a[(n + 16 - #)/16] /. HC /. HC], #]) &, Range[16]];
Part[FoldList[Mod[Plus[#1, #2], 4] &, 0,
  a[#] & /@ Range[4^4] /. {F[n_] :> 0, L[n_] :> 1, R[n_] :> -1}],
2 ;; -1] (* Bradley Klee, Aug 07 2015 *)

(define (A163540 n) (modulo (+ 3 (A163538 n) (A163539 n) (abs (A163539 n))) 4))

a(n) = A010873(A163538(n)+A163539(n)+abs(A163539(n))+3).

A261096 A(i,j) = rank (in A055089) of the composition of the i-th and the j-th permutation in table A055089, which lists all finite permutations in reversed colexicographic ordering.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 4, 3, 3, 4, 5, 0, 2, 4, 5, 2, 1, 5, 5, 5, 6, 3, 5, 4, 1, 4, 6, 7, 7, 4, 0, 0, 3, 7, 7, 8, 6, 12, 1, 3, 2, 8, 6, 8, 9, 10, 13, 13, 2, 1, 9, 10, 9, 9, 10, 11, 14, 12, 18, 0, 10, 11, 6, 8, 10, 11, 8, 15, 16, 19, 19, 11, 8, 7, 11, 11, 11, 12, 9, 16, 17, 20, 18, 0, 9, 11, 10, 7, 10, 12, 13, 18, 17, 14, 21, 22, 1, 1, 10, 6, 6, 9, 13, 13, 14, 19, 6, 15, 22, 23, 2, 0, 14, 7, 9, 8, 14, 12, 14

Offset: 0

Antti Karttunen, Aug 26 2015

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...
A(i,j) gives the rank (in ordering used by table A055089) of the permutation which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A055089 (note that the identity permutation is the 0th). Here the convention is that "permutations act of the left", thus, if p1 and p2 are permutations, then the product of p1 and p2 (p1 * p2) is defined such that (p1 * p2)(i) = p1(p2(i)) for i=1...
Each row and column is a permutation of A001477, because this is the Cayley table ("multiplication table") of an infinite enumerable group, namely, that subgroup of the infinite symmetric group (S_inf) which consists of permutations moving only finite number of elements.

			The top left corner of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  4,  5,  2,  3,  7,  6, 10, 11,  8,  9, 18, ...
   2,  3,  0,  1,  5,  4, 12, 13, 14, 15, 16, 17,  6, ...
   3,  2,  5,  4,  0,  1, 13, 12, 16, 17, 14, 15, 19, ...
   4,  5,  1,  0,  3,  2, 18, 19, 20, 21, 22, 23,  7, ...
   5,  4,  3,  2,  1,  0, 19, 18, 22, 23, 20, 21, 13, ...
   6,  7,  8,  9, 10, 11,  0,  1,  2,  3,  4,  5, 14, ...
   7,  6, 10, 11,  8,  9,  1,  0,  4,  5,  2,  3, 20, ...
   8,  9,  6,  7, 11, 10, 14, 15, 12, 13, 17, 16,  0, ...
   9,  8, 11, 10,  6,  7, 15, 14, 17, 16, 12, 13, 21, ...
  10, 11,  7,  6,  9,  8, 20, 21, 18, 19, 23, 22,  1, ...
  11, 10,  9,  8,  7,  6, 21, 20, 23, 22, 18, 19, 15, ...
  12, 13, 14, 15, 16, 17,  2,  3,  0,  1,  5,  4,  8, ...
  ...
For A(1,2) (row=1, column=2, both starting from zero), we take as permutation p the permutation which has rank=1 in the ordering used by A055089, which is a simple transposition (1 2), which we can extend with fixed terms as far as we wish (e.g., like {2,1,3,4,5,...}), and as permutation q we take the permutation which has rank=2 (in the same list), which is {1,3,2}. We compose these from the left, so that the latter one, q, acts first, thus c(i) = p(q(i)), and the result is permutation {2,3,1}, which is listed as the 4th one in A055089, thus A(1,2) = 4.
For A(2,1) we compose those two permutations in opposite order, as d(i) = q(p(i)), which gives permutation {3,1,2} which is listed as the 3rd one in A055089, thus A(2,1) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of array
Wikipedia, Cayley table

Transpose: A261097.
Row 0 & Column 0: A001477 (identity permutation).
Row 1: A261098.
Column 1: A004442.
Main diagonal: A261099.
Cf. tables A055089, A195663.
Cf. also A261216, A261217 (similar arrays, but using different orderings of permutations).
Permutations used in conjugation-formulas: A056019, A060119, A060120, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A056019(A261097(A056019(i),A056019(j))).
A(i,j) = A060119(A261216(A060126(i),A060126(j))).
A(i,j) = A060120(A261217(A060127(i),A060127(j))).

A261216 A(i,j) = rank (in A060117) of the composition of the i-th and the j-th permutation in table A060117, which lists all finite permutations.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 5, 3, 3, 4, 4, 0, 2, 4, 5, 3, 1, 4, 5, 5, 6, 2, 5, 5, 3, 4, 6, 7, 7, 4, 1, 2, 1, 7, 7, 8, 6, 14, 0, 0, 0, 8, 6, 8, 9, 11, 15, 15, 1, 2, 9, 11, 9, 9, 10, 10, 12, 14, 22, 3, 10, 10, 6, 8, 10, 11, 9, 13, 16, 23, 23, 11, 9, 7, 10, 11, 11, 12, 8, 17, 17, 21, 22, 0, 8, 11, 11, 9, 10, 12, 13, 19, 16, 13, 20, 19, 1, 1, 10, 7, 8, 7, 13, 13, 14, 18, 8, 12, 18, 18, 2, 0, 12, 6, 6, 6, 14, 12, 14

Offset: 0

Antti Karttunen, Aug 26 2015

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...
A(i,j) gives the rank of the permutation (in ordering used by table A060117) which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A060117 (note that the identity permutation is the 0th). Here the convention is that "permutations act of the left", thus, if p1 and p2 are permutations, then the product of p1 and p2 (p1 * p2) is defined such that (p1 * p2)(i) = p1(p2(i)) for i=1...
Equally, A(i,j) gives the rank in A060118 of the composition of the i-th and the j-th permutation in A060118, when convention is that "permutations act on the right".
Each row and column is a permutation of A001477, because this is the Cayley table ("multiplication table") of an infinite enumerable group, namely, that subgroup of the infinite symmetric group (S_inf) which consists of permutations moving only finite number of elements.

			The top left corner of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  5,  4,  3,  2,  7,  6, 11, 10,  9,  8, 19, ...
   2,  3,  0,  1,  5,  4, 14, 15, 12, 13, 17, 16,  8, ...
   3,  2,  4,  5,  1,  0, 15, 14, 16, 17, 13, 12, 21, ...
   4,  5,  3,  2,  0,  1, 22, 23, 21, 20, 18, 19, 16, ...
   5,  4,  1,  0,  2,  3, 23, 22, 19, 18, 20, 21, 11, ...
   6,  7,  8,  9, 10, 11,  0,  1,  2,  3,  4,  5, 14, ...
   7,  6, 11, 10,  9,  8,  1,  0,  5,  4,  3,  2, 23, ...
   8,  9,  6,  7, 11, 10, 12, 13, 14, 15, 16, 17,  2, ...
   9,  8, 10, 11,  7,  6, 13, 12, 17, 16, 15, 14, 20, ...
  10, 11,  9,  8,  6,  7, 18, 19, 20, 21, 22, 23, 17, ...
  11, 10,  7,  6,  8,  9, 19, 18, 23, 22, 21, 20,  5, ...
  12, 13, 14, 15, 16, 17,  8,  9,  6,  7, 11, 10,  0, ...
  ...
For A(1,2) (row=1, column=2, both starting from zero), we take as permutation p the permutation which has rank=1 in the ordering used by A060117, which is a simple transposition (1 2), which we can extend with fixed terms as far as we wish (e.g., like {2,1,3,4,5,...}), and as permutation q we take the permutation which has rank=2 (in the same list), which is {1,3,2}. We compose these from the left, so that the latter one, q, acts first, thus c(i) = p(q(i)), and the result is permutation {2,3,1}, which is listed as the 5th one in A060117, thus A(1,2) = 5.
For A(2,1) we compose those two permutations in opposite order, as d(i) = q(p(i)), which gives permutation {3,1,2} which is listed as the 3rd one in A060117, thus A(2,1) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..10439; the first 144 antidiagonals of array
Wikipedia, Cayley table

Transpose: A261217.
Row 0 & Column 0: A001477 (identity permutation).
Row 1: A261218.
Column 1: A004442.
Main diagonal: A261219.
Cf. also A060117, A060118, A261096, A261097.
Permutations used in conjugation-formulas: A060119, A060120, A060125, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A060125(A261217(A060125(i),A060125(j))).
A(i,j) = A060126(A261096(A060119(i),A060119(j))).
A(i,j) = A060127(A261097(A060120(i),A060120(j))).

A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.

Original entry on oeis.org

1, 1, 4, 6, 11, 15, 22, 28, 37, 45, 56, 66, 79, 91, 106, 120, 137, 153, 172, 190, 211, 231, 254, 276, 301, 325, 352, 378, 407, 435, 466, 496, 529, 561, 596, 630, 667, 703, 742, 780, 821, 861, 904, 946, 991, 1035, 1082, 1128, 1177, 1225, 1276, 1326, 1379, 1431

Offset: 0

Paul Barry, May 31 2003

Old name was "Modified triangular numbers".

Starting with offset 1 = row sums of an infinite lower triangular matrix with alternate columns of (1, 3, 5, 7, ...) and (1, 0, 0, 0, ...) (see example). - Gary W. Adamson, May 14 2010

			From _Gary W. Adamson_, May 14 2010: (Start)
First few rows of the triangle with row sums = A084263 =
1;
3, 1;
5, 0, 1;
7, 0, 3, 1;
9, 0, 5, 0, 1;
11, 0, 7, 0, 3, 1;
...
Example: a(4) = 11 = (7 + 0 + 3 + 1). (End)

Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

Partial sums of A004442.

Cf. A000217, A002522, A059841, A084264, A084570.

[(-1)^n/2+(n^2+n+1)/2 : n in [0..80]]; // Wesley Ivan Hurt, Dec 23 2021

Table[(-1)^n/2 + (n^2 + n + 1)/2, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 23 2021 *)
LinearRecurrence[{2,0,-2,1},{1,1,4,6},60] (* Harvey P. Dale, Jun 10 2023 *)

a(n)=n*(n+1)/2+(1-n)%2 \\ Charles R Greathouse IV, Jun 04 2013

E.g.f.: cosh(x)+exp(x)*(x+x^2/2).
a(n) = Sum_{k=0..n} k+(-1)^k.
a(n) = A000217(n)+A059841(n). Partial sums are A084570. Binomial transform is A084264.
G.f.: (1-x+2*x^2)/((1-x)^3*(1+x)). - R. J. Mathar, Apr 02 2008
a(0) = 1, a(n) = n^2 - a(n-1) + 1 for n >= 1. - Richard R. Forberg, Jun 05 2013
a(n) = 1 + floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
a(n) + a(n+1) = A002522(n+1). - R. J. Mathar, May 21 2018
a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). - Wesley Ivan Hurt, Dec 23 2021

Name changed by Wesley Ivan Hurt, Dec 23 2021

A126006 Involution of nonnegative integers: Swap the positions of digits q0 <-> q1, q2 <-> q3, q4 <-> q5, etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).

Original entry on oeis.org

0, 4, 8, 12, 1, 5, 9, 13, 2, 6, 10, 14, 3, 7, 11, 15, 64, 68, 72, 76, 65, 69, 73, 77, 66, 70, 74, 78, 67, 71, 75, 79, 128, 132, 136, 140, 129, 133, 137, 141, 130, 134, 138, 142, 131, 135, 139, 143, 192, 196, 200, 204, 193, 197, 201, 205, 194, 198, 202, 206, 195

Offset: 0

Antti Karttunen, Jan 02 2007

			29 = 0*64 + 1*16 + 3*4 + 1, i.e., 131 in quaternary and when digits are swapped in pairs, results 1013 in quaternary (1*64 + 0*16 + 1*4 + 3 = 71 in decimal), thus a(29)=71.

Index entries for sequences that are permutations of the natural numbers

Cf. A126007. A057300 is the analogous permutation based on swapping the binary digits of n.

Cf. A004442.

#include 
uint32_t a(uint32_t n) { return ((n & 0x33333333) << 2) | ((n & 0xcccccccc) >> 2); } /* Falk Hüffner, Jan 23 2022 */

a(n) = my(d=Vecrev(digits(n, 4))); if (#d % 2, d = concat(d, 0)); fromdigits(Vecrev(vector(#d, i, d[i+(-1)^(i-1)])), 4); \\ Michel Marcus, Jan 23 2022

(define (A126006 n) (let loop ((n n) (s 0) (p 1)) (cond ((zero? n) s) (else (loop (floor->exact (/ n 16)) (+ s (* p (+ (* 4 (modulo n 4)) (modulo (floor->exact (/ n 4)) 4)))) (* p 16))))))

A163534 The absolute direction (0=east, 1=south, 2=west, 3=north) of the Peano curve A163334 at point n.

Original entry on oeis.org

0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 3, 2, 2, 3, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 2, 2, 2, 3, 0, 0, 3, 2, 2, 2, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 3, 2, 2, 3, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 3, 2, 2, 3, 0, 0, 0, 0, 0, 1, 2, 2, 1, 0, 0, 0, 0, 0, 3

Offset: 1

Antti Karttunen, Aug 01 2009

nonn

Taking every ninth term gives the same sequence: (and similarly for all higher powers of 9 as well): a(n) = a(9*n).

Antti Karttunen, Table of n, a(n) for n = 1..59049

Cf. A163535 (transposed direction), A163536 (turn).

(define (A163534 n) (modulo (+ 3 (A163532 n) (A163533 n) (abs (A163533 n))) 4))

a(n) = A010873(A163532(n)+A163533(n)+abs(A163533(n))+3).

a(n) = A004442(A163535(n)).

Name corrected by Kevin Ryde, Aug 29 2020

A261097 Transpose of square array A261096.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 3, 4, 3, 4, 2, 0, 5, 4, 5, 5, 5, 1, 2, 5, 6, 4, 1, 4, 5, 3, 6, 7, 7, 3, 0, 0, 4, 7, 7, 8, 6, 8, 2, 3, 1, 12, 6, 8, 9, 9, 10, 9, 1, 2, 13, 13, 10, 9, 10, 8, 6, 11, 10, 0, 18, 12, 14, 11, 10, 11, 11, 11, 7, 8, 11, 19, 19, 16, 15, 8, 11, 12, 10, 7, 10, 11, 9, 0, 18, 20, 17, 16, 9, 12, 13, 13, 9, 6, 6, 10, 1, 1, 22, 21, 14, 17, 18, 13, 14, 12, 14, 8, 9, 7, 14, 0, 2, 23, 22, 15, 6, 19, 14

Offset: 0

Antti Karttunen, Aug 26 2015

Each row and column is a permutation of A001477. See the comments at A261096.

			The top left corner of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, ...
   2,  4,  0,  5,  1,  3,  8, 10,  6, 11,  7,  9, 14, ...
   3,  5,  1,  4,  0,  2,  9, 11,  7, 10,  6,  8, 15, ...
   4,  2,  5,  0,  3,  1, 10,  8, 11,  6,  9,  7, 16, ...
   5,  3,  4,  1,  2,  0, 11,  9, 10,  7,  8,  6, 17, ...
   6,  7, 12, 13, 18, 19,  0,  1, 14, 15, 20, 21,  2, ...
   7,  6, 13, 12, 19, 18,  1,  0, 15, 14, 21, 20,  3, ...
   8, 10, 14, 16, 20, 22,  2,  4, 12, 17, 18, 23,  0, ...
   9, 11, 15, 17, 21, 23,  3,  5, 13, 16, 19, 22,  1, ...
  10,  8, 16, 14, 22, 20,  4,  2, 17, 12, 23, 18,  5, ...
  11,  9, 17, 15, 23, 21,  5,  3, 16, 13, 22, 19,  4, ...
  12, 18,  6, 19,  7, 13, 14, 20,  0, 21,  1, 15,  8, ...
  ...

Antti Karttunen, Table of n, a(n) for n = 0..7259; the first 120 antidiagonals of array
Wikipedia, Cayley table

Transpose: A261096.
Row 0 & Column 0: A001477 (identity permutation).
Row 1: A004442.
Column 1: A261098.
Main diagonal: A261099.
Cf. also A055089, A195663.
Cf. also A261216, A261217 (similar arrays, but using different orderings of permutations).
Permutations used in conjugation-formulas: A056019, A060119, A060120, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A056019(A261096(A056019(i),A056019(j))).
A(i,j) = A060119(A261217(A060126(i),A060126(j))).
A(i,j) = A060120(A261216(A060127(i),A060127(j))).

A261217 A(i,j) = rank (in A060118) of the composition of the i-th and the j-th permutation in table A060118, which lists all finite permutations.

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 3, 5, 3, 4, 2, 0, 4, 4, 5, 5, 4, 1, 3, 5, 6, 4, 3, 5, 5, 2, 6, 7, 7, 1, 2, 1, 4, 7, 7, 8, 6, 8, 0, 0, 0, 14, 6, 8, 9, 9, 11, 9, 2, 1, 15, 15, 11, 9, 10, 8, 6, 10, 10, 3, 22, 14, 12, 10, 10, 11, 11, 10, 7, 9, 11, 23, 23, 16, 13, 9, 11, 12, 10, 9, 11, 11, 8, 0, 22, 21, 17, 17, 8, 12, 13, 13, 7, 8, 7, 10, 1, 1, 19, 20, 13, 16, 19, 13, 14, 12, 14, 6, 6, 6, 12, 0, 2, 18, 18, 12, 8, 18, 14

Offset: 0

Antti Karttunen, Aug 26 2015

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...
A(i,j) gives the rank (in ordering used by table A060118) of the permutation which is obtained by composing permutations p and q listed as the i-th and the j-th permutation in irregular table A060118 (note that the identity permutation is the 0th). Here the convention is that "permutations act of the left", thus, if p1 and p2 are permutations, then the product of p1 and p2 (p1 * p2) is defined such that (p1 * p2)(i) = p1(p2(i)) for i=1...
Equally, A(i,j) gives the rank in A060117 of the composition of the i-th and the j-th permutation in A060117, when convention is that "permutations act on the right".
Each row and column is a permutation of A001477, because this is the Cayley table ("multiplication table") of an infinite enumerable group, namely, that subgroup of the infinite symmetric group (S_inf) which consists of permutations moving only finite number of elements.

			The top left corner of the array:
   0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, ...
   1,  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, ...
   2,  5,  0,  4,  3,  1,  8, 11,  6, 10,  9,  7, 14, ...
   3,  4,  1,  5,  2,  0,  9, 10,  7, 11,  8,  6, 15, ...
   4,  3,  5,  1,  0,  2, 10,  9, 11,  7,  6,  8, 16, ...
   5,  2,  4,  0,  1,  3, 11,  8, 10,  6,  7,  9, 17, ...
   6,  7, 14, 15, 22, 23,  0,  1, 12, 13, 18, 19,  8, ...
   7,  6, 15, 14, 23, 22,  1,  0, 13, 12, 19, 18,  9, ...
   8, 11, 12, 16, 21, 19,  2,  5, 14, 17, 20, 23,  6, ...
   9, 10, 13, 17, 20, 18,  3,  4, 15, 16, 21, 22,  7, ...
  10,  9, 17, 13, 18, 20,  4,  3, 16, 15, 22, 21, 11, ...
  11,  8, 16, 12, 19, 21,  5,  2, 17, 14, 23, 20, 10, ...
  12, 19,  8, 21, 16, 11, 14, 23,  2, 20, 17,  5,  0, ...
  ...
For A(1,2) (row=1, column=2, both starting from zero), we take as permutation p the permutation which has rank=1 in the ordering used by A060118, which is a simple transposition (1 2), which we can extend with fixed terms as far as we wish (e.g., like {2,1,3,4,5,...}), and as permutation q we take the permutation which has rank=2 (in the same list), which is {1,3,2}. We compose these from the left, so that the latter one, q, acts first, thus c(i) = p(q(i)), and the result is permutation {2,3,1}, which is listed as the 3rd one in A060118, thus A(1,2) = 3.
For A(2,1) we compose those two permutations in opposite order, as d(i) = q(p(i)), which gives permutation {3,1,2} which is listed as the 5th one in A060118, thus A(2,1) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..7259; the first 120 antidiagonals of array
Wikipedia, Cayley table

Transpose: A261216.
Row 0 & Column 0: A001477 (identity permutation)
Row 1: A004442.
Column 1: A261218.
Main diagonal: A261219.
Cf. also A060117, A060118, A261096, A261097.
Cf. also A089839.
Permutations used in conjugation-formulas: A060119, A060120, A060125, A060126, A060127.

By conjugating with related permutations and arrays:
A(i,j) = A060125(A261216(A060125(i),A060125(j))).
A(i,j) = A060127(A261096(A060120(i),A060120(j))).
A(i,j) = A060126(A261097(A060119(i),A060119(j))).

cp's OEIS Frontend

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

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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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A163540 The absolute direction (0=east, 1=south, 2=west, 3=north) taken by the type I Hilbert's Hamiltonian walk A163357 at the step n.

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A261096 A(i,j) = rank (in A055089) of the composition of the i-th and the j-th permutation in table A055089, which lists all finite permutations in reversed colexicographic ordering.

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A261216 A(i,j) = rank (in A060117) of the composition of the i-th and the j-th permutation in table A060117, which lists all finite permutations.

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A084263 a(n) = (-1)^n/2+(n^2+n+1)/2.

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A126006 Involution of nonnegative integers: Swap the positions of digits q0 <-> q1, q2 <-> q3, q4 <-> q5, etc. in the base-4 expansion of n (where n = ... + q4*256 + q3*64 + q2*16 + q1*4 + q0).

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C

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A126006 Involution of nonnegative integers: Swap the positions of digits q0 <-> q1, q2 <-> q3, q4 <-> q5, etc. in the base-4 expansion of n (where n = ... + q4256 + q364 + q216 + q14 + q0).