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

Author

Tilman Piesk, Aug 10 2025

Keywords

• nonn
• new

Comments

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 |

+-----------------------+---------+

Examples

			  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.
		

Links

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

Formula

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

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