A387130 -id:A387130 - 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.

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

Original entry on oeis.org

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).

cp's OEIS Frontend

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

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