A383904 -id:A383904 - cp's OEIS frontend

A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.

0, 0, 1, 1, 2, 1, 5, 1, 10, 4, 27, 1, 88, 1, 247, 29, 810, 1, 2780, 1, 9260, 249, 32067, 1, 113520, 26, 400025, 2704, 1432868, 1, 5179905, 1, 18784170, 32069, 68635479, 271, 252201136, 1, 930138523, 400027, 3446168660, 1, 12817096533, 1, 47820447036, 5173304

Offset: 0

Tilman Piesk, Aug 10 2025

A003239(n) is the number of 2n-bead balanced binary necklaces. A022553(n) among them are primitive.
The remaining a(n) necklaces are periodic.
Sequences counting 2n-bead balanced binary necklaces:
                       primitive  imprimitive
                     +-----------------------+---------+
  self-complementary |  A000048     A115118  | A000013 |
   complement pairs  |  A383904     A387130  | A386388 |
                     +-----------------------+---------+
                     |  A022553      this    | A003239 |
                     +-----------------------+---------+

			  n | A003239(n) A022553(n) | a(n)
  0 |         1          1  |   0
  1 |         1          1  |   0
  2 |         2          1  |   1
  3 |         4          3  |   1
  4 |        10          8  |   2
  5 |        26         25  |   1
  6 |        80         75  |   5
  7 |       246        245  |   1
  8 |       810        800  |  10
  9 |      2704       2700  |   4
 10 |      9252       9225  |  27
 11 |     32066      32065  |   1
 12 |    112720     112632  |  88
 13 |    400024     400023  |   1
 14 |   1432860    1432613  | 247
 15 |   5170604    5170575  |  29
 16 |  18784170   18783360  | 810
There are A003239(8) = 810 balanced binary necklaces of length 16. A022553(8) = 800 of them are primitive. a(n) = 10 are not. See A387130 for a list.

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

a(n) = A003239(n) - A022553(n).

a(n) = A115118(n) + 2 * A387130(n).

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

A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.

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