A056023 -id:A056023 - cp's OEIS frontend

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

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

Offset: 1

Clark Kimberling, Aug 01 2000

A triangle such that (1) every positive integer occurs exactly once; (2) row n consists of n consecutive numbers; (3) odd-numbered rows are increasing; and (4) even-numbered rows are decreasing.
Self-inverse permutation of the natural numbers.
Mirror image of triangle in A056023. - Philippe Deléham, Apr 04 2009
a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers. - Boris Putievskiy, Dec 24 2012
For generalizations see A218890, A213927. - Boris Putievskiy, Mar 10 2013

			The start of the sequence as a table:
   1,  3,  4, 10, 11, 21, ...
   2,  5,  9, 12, 20, 23, ...
   6,  8, 13, 19, 24, 34, ...
   7, 14, 18, 25, 33, 40, ...
  15, 17, 26, 32, 41, 51, ...
  ...
Enumeration by boustrophedonic ("ox-plowing") diagonal method. - _Boris Putievskiy_, Dec 24 2012
The start of the sequence as triangle array read by rows:
   1;
   3,  2;
   4,  5,  6;
  10,  9,  8,  7;
  11, 12, 13, 14, 15;
  ...

Reinhard Zumkeller, Rows n = 1..125 of table, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions, arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers

Cf. A079826, A131179 (first column), A218890, A213927.

Cf. A000027, A038722.

a056011 n = a056011_tabl !! (n-1)
a056011_list = concat a056011_tabl
a056011_tabl = ox False a000027_tabl where
  ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss
a056011_row n = a056011_tabl !! (n-1)
-- Reinhard Zumkeller, Nov 08 2013

A056011 := proc(n,k)
        if type(n,'even') then
                A131179(n)-k+1 ;
        else
                A131179(n)+k-1 ;
        end if;
end proc: # R. J. Mathar, Sep 05 2012

Flatten[If[EvenQ[Length[#]],Reverse[#],#]&/@Table[c=(n(n+1))/2;Range[ c-n+1,c],{n,20}]] (* Harvey P. Dale, Mar 25 2012 *)
With[{nn=20},{#[[1]],Reverse[#[[2]]]}&/@Partition[ TakeList[ Range[ (nn(nn+1))/2],Range[nn]],2]//Flatten] (* Harvey P. Dale, Oct 05 2021 *)

a(n) = ((i+j-1)*(i+j-2)+((-1)^t+1)*i - ((-1)^t-1)*j)/2, where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 24 2012

New name from Peter Luschny, Apr 15 2023, based on Boris Putievskiy's comment

A376214 a(n) = A379342(A379343(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Feb 05 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. For n > 0, the length of row n is 4n-3 = A016813(n-1).
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A379343 and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A000027 (the identity permutation), A381662, A380817, A379343, A380200, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Apr 27 2025
A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A382679, A376217, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 27 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A382679, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, 383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 27 2025

			Triangle array begins:
  k=    1   2  3  4   5   6   7   8   9
  n=1:  1;
  n=2:  2,  3, 4, 5,  6;
  n=3:  7, 10, 9, 8, 11, 14, 13, 12, 15;
(1, 2, 3, ..., 12, 15) = (1, 2, 3, ..., 12, 15)^(-1).
(1, 2, 3, ..., 12, 15) = (1, 5, 2, ..., 10, 15) (1, 3, 5, ..., 7, 15). The first permutation is from Example A380245 and the second from Example A378684.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6,  8, 15, ...
   2,  5,  9, 12, 20, ...
   4, 10, 13, 19, 26, ...
   7, 14, 18, 25, 33, ...
  11, 21, 24, 34, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1,
  1, 2, 3, 4, 5;
  1, 4, 3, 2, 5, 8, 7, 6, 9.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Alternating Group A4: Subgroups and the Cayley Table (2025).
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table (2025 D4xC2).
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

a(n) = A380245(A378684(n)).

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n,k) = k if k < m and k == 1 (mod 2), P(n,k) = -k + m + 1 if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n,k) = -k + 3m - 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.

Name corrected by Pontus von Brömssen, Jun 24 2025

A380817 a(n) = A379343(A380245(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Feb 04 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n>1, row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A379343 and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A000027 (the identity permutation), A381662, A376214, A379343, A380200, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Apr 17 2025
This sequence and A381968 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - Boris Putievskiy, Jun 09 2025
This sequence, A378762 and A381968 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - Boris Putievskiy, Jun 09 2025

			Triangle array begins:
  k=    1   2   3  4   5   6   7   8   9
  n=1:  1;
  n=2:  2,  3,  4, 5,  6;
  n=3:  9, 10,  7, 8, 11, 12, 13, 14, 15;
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6,  8, 15, ...
   2,  5,  7, 14, 16, ...
   4, 10, 13, 19, 26, ...
   9, 12, 18, 25, 31, ...
  11, 21, 24, 34, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1,
  1, 2, 3, 4, 5;
  3, 4, 1, 2, 5, 6, 7, 8, 9.
All permutations are self-inverse.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Alternating Group A4: Subgroups and the Cayley Table (2025).
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

For 1 <= k <= 4n - 3, T(n,k) = A000384(n-1) + P(n,k), P(n,k) = -k + m - 1 if k < m and k == 1 (mod 2), P(n,k) = -k + m + 1 if k < m and k == 0 (mod 2), P(n,k) = k if k >= m, where m = 2*n - 1.

Name corrected by Pontus von Brömssen, Jun 24 2025

A381662 a(n) = A378684(A380200(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 03 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A379343 and A378684 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the alternating group A4. The list of the 12 elements of that group: this sequence, A000027 (the identity permutation), A380817, A376214, A379343, A380200, A378684, A379342, A380245, A381664, A380815, A381663. For subgroups and the Cayley table of the group A4 see Boris Putievskiy (2025) link. - Boris Putievskiy, Apr 28 2025
A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A000027 (the identity permutation), A381968, A382499, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 28 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A382499, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 21 2025

			Triangle array begins:
  k=    1  2  3   4   5   6   7   8   9
  n=1:  1;
  n=2:  2, 3, 4,  5,  6;
  n=3:  9, 8, 7, 10, 11, 14, 13, 12, 15;
(1,2,3,...,12,15) = (1,2,3,...,12,15)^(-1).
(1,2,3,...,12,15) = (1,5,2,...,8,15) (1,3,5,...,9,15). The first permutation is from Example A379342 and the second from Example A379343.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6, 10, 15, ...
   2,  5,  7, 12, 16, ...
   4,  8, 13, 19, 26, ...
   9, 14, 18, 25, 31, ...
  11, 17, 24, 32, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  1, 2, 3, 4, 5;
  3, 2, 1, 4, 5, 8, 7, 6, 9.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Alternating Group A4: Subgroups and the Cayley Table (2025).
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table(2025 D4xC2).
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Alternating Group.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

a(n) = A379342(A379343(n)) = A381968(A381968(n)).

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n,k) = - k + m - 1 if k < m and k == 1 (mod 2), P(n,k) = k if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n,k) = -k + 3 * m - 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.

Name corrected by Pontus von Brömssen, Jun 24 2025

A378762 a(n) = A378200(A378200(A378200(n))).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Dec 06 2024

The sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n>1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of natural numbers.
The sequence is an intra-block permutation of the positive integers.
Generalization of Cantor numbering method.
The sequence A378200 generates the cyclic group C6 under composition. The elements of C6 are the successive compositions of A378200 with itself: A378684(n) = A378200(A378200(n)) = A378200^2(n), a(n) = A378200^3(n), A379342(n) = A378200^4(n), A378705(n) = A378200^5(n). The identity element is A000027(n) = A378200^6(n). - Boris Putievskiy, Jan 15 2025
This sequence, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A382679, A376214, A382680, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - Boris Putievskiy, Jun 09 2025

			Triangle array begins:
  k=   1   2   3   4   5   6   7   8   9
  n=1: 1;
  n=2: 2,  3,  6,  5,  4;
  n=3: 9, 10,  7,  8, 15, 14, 13, 12, 11;
(1, 2, 3, ..., 12, 11) = (A378200(1), A378200(2), A378200(3), ..., A378200(14), A378200(15))^3.
(1, 2, 3, ..., 12, 11) = (1, 2, 3, ..., 12, 11)^(-1).
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  4,  8, 11, ...
   2,  5,  7, 12, 16, ...
   6, 10, 13, 19, 24, ...
   9, 14, 18, 25, 31, ...
  15, 21, 26, 34, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  1, 2, 5, 4, 3;
  3, 4, 1, 2, 9, 8, 7, 6, 5.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

Mathematica
```
P[n_,k_]:=Module[{m=2*n-1},If[k
				
```

Linear sequence: (a(1), a(2), ..., a(A000384(n+1))) is a permutation of the positive integers from 1 to A000384(n+1). (a(1), a(2), ..., a(A000384(n+1))) = (A378200(1), A378200(2), ..., A378200(A000384(n+1)))^3. (a(1), a(2), ..., a(A000384(n+1))) = (a(1), a(2), ..., a(A000384(n+1)))^(-1).

Triangular array T(n,k) for 1 <= k <= 4n - 3 (see Example): T(n,k) = A000384(n-1) + P(n,k), P(n,k) = m - k - 1 if k < m and k == 1 (mod 2), P(n, k) = m - k + 1 if k < m and k == 0 (mod 2), P(n, k) = 3m - k - 1 if k >= m, where m = 2n - 1.

A381968 a(a(n)) = A381662(n).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 12 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
This sequence and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A382499 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - Boris Putievskiy, Jun 09 2025
This sequence,  A378762 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A382499 (the inverse permutation), A381662, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - Boris Putievskiy, Jun 09 2025

			Triangle array begins:
  k=     1  2   3   4   5   6   7   8   9
  n=1:   1;
  n=2:   5, 3,  4,  2,  6;
  n=3:  14, 8, 12, 10, 11,  7, 13,  9, 15;
(1,5,3,...,9,15) (1,5,3,...,9,15) = (1,2,3,...,12,15). The permutation on the right-hand side is from Example A381662.
ord(1,5,3,...,9,15) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6, 10, 15, ...
   5,  2, 12,  9, 23, ...
   4,  8, 13, 19, 26, ...
  14,  7, 25, 18, 40, ...
  11, 17, 24, 32, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 2, 3, 1, 5;
  8, 2, 6, 4, 5, 1, 7, 3, 9.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A382499, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n,k) = - k + 2m - 1 if k < m and k == 1 (mod 2), P(n,k) = k if k < m and k == 0 (mod 2), P(n,k) = k if k >= m and k == 1 (mod 2), P(n,k) = k - m if k >= m and k == 0 (mod 2), where m = 2n - 1.

A382499 Inverse permutation to A381968.

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Mar 29 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A381968 and  and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral group D4. The list of the 8 elements of that group: this sequence, A381968 (the inverse permutation), A000027 (the identity permutation), A381662, A380817, A382679, A376214, A382680. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 17 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968 (the inverse permutation), A381662, A380817, A382679, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 27 2025

			Triangle array begins:
  k=     1  2   3   4   5  6   7  8   9
  n=1:   1;
  n=2:   5, 3,  4,  2,  6;
  n=3:  12, 8, 14, 10, 11, 9, 13, 7, 15;
(1,5,3,...,7,15)^(-1) = (1,5,3,...,9,15). The permutation on the right-hand side is from Example A381968.
ord(1,5,3,...,7,15) = 4.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6, 10, 15, ...
   5,  2, 14,  7, 27, ...
   4,  8, 13, 19, 26, ...
  12,  9, 25, 18, 42, ...
  11, 17, 24, 32, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 2, 3, 1, 5;
  6, 2, 8, 4, 5, 3, 7, 1, 9.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table (2025 D4xC2).
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

ord(a(1), a(2), ..., a(A000384(n+1))) = 4, where ord is the order of the permutation.

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = k + m if k < m and k == 1 (mod 2), P(n, k) = k if k < m and k == 0 (mod 2), P(n, k) = k if k >= m and k == 1 (mod 2), P(n, k) = -k + 2m - 1 if k >= m and k == 0 (mod 2), where m = 2n - 1.

A382679 a(n) = A381968(A380817(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Apr 03 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A381968 and  and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral Group D4. The list of the 8 elements of that group: this sequence, A382680 (the inverse permutation), A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214. For subgroups and the Cayley table of the group D4 see Boris Putievskiy (2025 D4 (I)) link. - Boris Putievskiy, Apr 27 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214, A382680, A378762, A383419, A383589, A383590, A056023, A383722, A383723, 383724. For subgroups and the Cayley table of the group D4xC2 see Boris Putievskiy (2025 D4xC2) link. - Boris Putievskiy, May 27 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  4, 2,  6;
  n=3: 14, 10, 12, 8, 11, 9, 13, 7, 15;
(1,5,3,...,7,15) = (1,5,3,...,7,15)^(-1).
(1,5,3,...,7,15) = (1,5,3,...,9,15) (1,2,3,...,10,7,...,14,15).  The first permutation on the right-hand side is from Example A381968 and the second from Example A380817.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
   1,  3,  6,  8, 15, ...
   5,  2, 12,  7, 23, ...
   4, 10, 13, 19, 26, ...
  14,  9, 25, 18, 40, ...
  11, 21, 24, 34, 41, ...
  ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1;
  4, 2, 3, 1, 5;
  8, 4, 6, 2, 5, 3, 7, 1, 9.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table (2025 D4 (I)).
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table (2025 D4xC2).
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382679, A382680, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = 2m - 1 - k if k < m and k == 1 (mod 2), P(n, k) = m + 1 - k if k < m and k == 0 (mod 2), P(n, k) = k if k >= m and k == 1 (mod 2), P(n, k) = 2m - 1 - k if k >= m and k == 0 (mod 2),

where m = 2n - 1.

A382680 a(n) = A382499(A380817(n)).

Original entry on oeis.org

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

Offset: 1

Boris Putievskiy, Apr 03 2025

This sequence can be regarded as a triangular array read by rows. Each row is a permutation of a block of consecutive numbers; the blocks are disjoint and every positive number belongs to some block. The length of row n is 4n-3 = A016813(n-1), n > 0.
The sequence can also be regarded as a table read by upward antidiagonals. For n > 1 row n joins two consecutive antidiagonals.
The sequence is a self-inverse permutation of the positive integers.
In particular, the initial {a(1), a(2), ..., a(A000384(n+1))} is self-inverse.
The sequence is an intra-block permutation of the positive integers.
Generalization of the Cantor numbering method.
A381968 and  and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the dihedral Group D4. The list of the 8 elements of that group: this sequence, A382679 (the inverse permutation), A000027 (the identity permutation), A381968, A381662, A382499, A380817, A376214. For subgroups and the Cayley table of the group D4 see Putievskiy (D4 (I)) link. - Boris Putievskiy, Apr 27 2025
A378762, A381968 and A380817 generate via composition a finite non-abelian group of permutations of positive integers, isomorphic to the direct product of the dihedral group D4 and the cyclic group C2. The list of the 16 elements of that group: this sequence, A000027 (the identity permutation), A381968, A381662, A382499, A380817, A382679, A376214, A378762, A383419, A383589, A383590, A056023, A383722, A383723, A383724. For subgroups and the Cayley table of the group D4xC2 see Putievskiy link. - Boris Putievskiy, Jun 02 2025

			Triangle array begins:
  k=    1   2   3  4   5  6   7  8   9
  n=1:  1;
  n=2:  5,  3,  4, 2,  6;
  n=3: 12, 10, 14, 8, 11, 7, 13, 9, 15;
(1,5,3,...,9,15) = (1,5,3,...,9,15)^(-1).
(1,5,3,...,9,15) = (1,5,3,...,7,15) (1,2,3,...,10,7,...,14,15). The first permutation on the right-hand side is from Example A382499 and the second from Example A380817.
For n > 1, each row of triangle array joins two consecutive upward antidiagonals in the table:
    1,  3,  6,  8, 15, ...
    5,  2, 14,  9, 27, ...
    4, 10, 13, 19, 26, ...
   12,  7, 25, 18, 42, ...
   11, 21, 24, 34, 41, ...
   ...
Subtracting (n-1)*(2*n-3) from each term in row n produces a permutation of numbers from 1 to 4*n-3:
  1,
  4, 2, 3, 1, 5,
  6, 4, 8, 2, 5, 1, 7, 3, 9.

Boris Putievskiy, Table of n, a(n) for n = 1..9730
Boris Putievskiy, Integer Sequences: Irregular Arrays and Intra-Block Permutations, arXiv:2310.18466 [math.CO], 2023.
Boris Putievskiy, The Dihedral Group D4 (I): Subgroups and the Cayley Table.
Boris Putievskiy, The Direct Product D4xC2: Subgroups and the Cayley Table.
Groupprops, Subgroup structure of direct product of D8 and Z2.
Eric Weisstein's World of Mathematics, Dihedral Group D_4.
Index entries for sequences that are permutations of the natural numbers.

Cf. A000027, A000384, A016813 (row lengths), A056023, A376214, A378684, A378762, A379342, A379343, A380200, A380245, A380815, A380817, A381662, A381663, A381664, A381968, A382499, A382679, A383419, A383589, A383590, A383722, A383723, A383724.

T[n_,k_]:=(n-1)*(2*n-3)+Module[{m=2*n-1},If[k

T(n,k) for 1 <= k <= 4n - 3: T(n,k) = A000384(n-1) + P(n,k), P(n, k) = m + k if k < m and k == 1 (mod 2), P(n, k) = m - k + 1 if k < m and k == 0 (mod 2), P(n, k) = k if k >= m and k == 1 (mod 2), P(n, k) = - m + k if k >= m and k == 0 (mod 2), where m = 2n - 1.

A373662 a(n) = (1 + (n+1)^2 - (n-2)*(-1)^n)/2.

Original entry on oeis.org

2, 5, 9, 12, 20, 23, 35, 38, 54, 57, 77, 80, 104, 107, 135, 138, 170, 173, 209, 212, 252, 255, 299, 302, 350, 353, 405, 408, 464, 467, 527, 530, 594, 597, 665, 668, 740, 743, 819, 822, 902, 905, 989, 992, 1080, 1083, 1175, 1178, 1274, 1277, 1377, 1380, 1484, 1487, 1595

Offset: 1

Wesley Ivan Hurt, Jun 12 2024

Fill an array with the natural numbers n = 1,2,... along diagonals in alternating 'down' and 'up' directions. a(n) is row 2 of the boustrophedon-style array (see example).

In general, row k is given by (1+t^2+(n-k)*(-1)^t)/2, t = n+k-1. Here, k=2.

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

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

For rows k = 1..10: A131179 (k=1) n>0, this sequence (k=2), A373663 (k=3), A374004 (k=4), A374005 (k=5), A374007 (k=6), A374008 (k=7), A374009 (k=8), A374010 (k=9), A374011 (k=10).
Cf. A001844, A038722.
Row n=2 of A056011.
Column k=2 of A056023.

[(1 + (n+1)^2 - (n-2)*(-1)^n)/2: n in [1..80]];

k := 2; Table[(1 + (n+k-1)^2 + (n-k) (-1)^(n+k-1))/2, {n, 80}]

def A373662(n): return ((n+1)*(n+2)-1 if n&1 else n*(n+1)+5)>>1 # Chai Wah Wu, Jun 23 2024

a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).
a(n) = A131179(n+1) + (-1)^n.
G.f.: -x*(2*x^4-3*x^3+3*x+2)/((x+1)^2*(x-1)^3). - Alois P. Heinz, Jun 12 2024

cp's OEIS Frontend

A056011 Enumeration of natural numbers by the boustrophedonic diagonal method.

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A376214 a(n) = A379342(A379343(n)).

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A380817 a(n) = A379343(A380245(n)).

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A381662 a(n) = A378684(A380200(n)).

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A378762 a(n) = A378200(A378200(A378200(n))).

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A381968 a(a(n)) = A381662(n).

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A382499 Inverse permutation to A381968.

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