A386946 -id:A386946 - cp's OEIS frontend

A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

0, 0, 0, 1, 3, 11, 35, 118, 392, 1336, 4587, 15986, 56231, 199854, 716014, 2584742, 9390656, 34315811, 126039218, 465062362, 1723066193, 6407806833, 23910159818, 89493721076, 335912335304, 1264105728831, 4768446686910, 18027215660947, 68291877609003

Offset: 0

Tilman Piesk, Aug 07 2025

A022553(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A000048 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 2, 6, 22, 70, 236, 784, 2672, 9174, 31972, 112462, 399708, 1432028, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |   this       A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A022553(n) A000048(n) | 2*a(n)    a(n)
  0 |         1          1  |     0       0
  1 |         1          1  |     0       0
  2 |         1          1  |     0       0
  3 |         3          1  |     2       1
  4 |         8          2  |     6       3
  5 |        25          3  |    22      11
  6 |        75          5  |    70      35
  7 |       245          9  |   236     118
  8 |       800         16  |   784     392
  9 |      2700         28  |  2672    1336
 10 |      9225         51  |  9174    4587
Examples for n=5 with necklaces of length 10:
The total number of necklaces is A003239(5) = 26.
Only A386946(5) = 1 of them is periodic, namely 0101010101.
The other A022553(5) = 25 are primitive.
A000048(5) = 3 among those are self-complementary:
 0000011111
 0001011101
 0010011011
The remaining 22 necklaces form a(5) = 11 complement pairs:
 0000101111 0000111101
 0000110111 0001111001
 0000111011 0001001111
 0001010111 0001110101
 0001011011 0010011101
 0001100111 0001110011
 0001101011 0010100111
 0001101101 0010010111
 0010101011 0011010101
 0010101101 0010110101
 0010110011 0011001101

Tilman Piesk, Table of n, a(n) for n = 0..1000

a(n) = (A022553(n) - A000048(n)) / 2.

A387130 a(n) is the number of complement pairs of imprimitive (periodic) 2n-bead balanced binary necklaces.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 3, 1, 11, 0, 39, 0, 118, 12, 395, 0, 1372, 0, 4601, 119, 15986, 0, 56662, 11, 199854, 1337, 716135, 0, 2589376, 0, 9391051, 15987, 34315811, 129, 126096824, 0, 465062362, 199855, 1723071186, 0, 6408523001, 0, 23910175807, 2586090, 89493721076

Offset: 0

Tilman Piesk, Aug 17 2025

A386946(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A115118 is the number of those that are self-complementary (i.e., can be rotated so that all beads change color). Their difference 2*a(n) is the number of those that are not self-complementary. a(n) is the number pairs of distinct complements.
Doubled entries: 0, 0, 0, 0, 0, 0, 2, 0, 6, 2, 22, 0, 78, 0, 236, 24, 790, 0, 2744, ...
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |  A383904      this    | A386388 |
                     +-----------------------+---------+
                     |  A022553     A386946  | A003239 |
                     +-----------------------+---------+

			  n | A386946(n) A115118(n) | 2*a(n)    a(n) | A386388(n) A383904(n)
  0 |         0          0  |     0       0  |         0          0
  1 |         0          0  |     0       0  |         0          0
  2 |         1          1  |     0       0  |         0          0
  3 |         1          1  |     0       0  |         1          1
  4 |         2          2  |     0       0  |         3          3
  5 |         1          1  |     0       0  |        11         11
  6 |         5          3  |     2       1  |        36         35
  7 |         1          1  |     0       0  |       118        118
  8 |        10          4  |     6       3  |       395        392
  9 |         4          2  |     2       1  |      1337       1336
 10 |        27          5  |    22      11  |      4598       4587
 11 |         1          1  |     0       0  |     15986      15986
 12 |        88         10  |    78      39  |     56270      56231
 13 |         1          1  |     0       0  |    199854     199854
 14 |       247         11  |   236     118  |    716132     716014
 15 |        29          5  |    24      12  |   2584754    2584742
 16 |       810         20  |   790     395  |   9391051    9390656
Examples for n=8 with necklaces of length 16:
The total number of necklaces is A003239(8) = 810.
A022553(8) = 800 of them are primitive.
The other A386946(8) = 10 are periodic.
A115118(8) = 4 among those are self-complementary:
 0000111100001111
 0010110100101101
 0011001100110011
 0101010101010101
The remaining 6 necklaces form a(8) = 3 complement pairs:
 0001011100010111 0001110100011101
 0001101100011011 0010011100100111
 0010101100101011 0011010100110101

Tilman Piesk, Table of n, a(n) for n = 0..1000

Cf. A386946, A115118, A386388, A383904.

a(n) = (A386946(n) - A115118(n)) / 2.

a(n) = A386388(n) - A383904(n).

cp's OEIS Frontend

A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.

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