A386946 - OEIS

%I #15 Aug 27 2025 18:20:26

%S 0,0,1,1,2,1,5,1,10,4,27,1,88,1,247,29,810,1,2780,1,9260,249,32067,1,

%T 113520,26,400025,2704,1432868,1,5179905,1,18784170,32069,68635479,

%U 271,252201136,1,930138523,400027,3446168660,1,12817096533,1,47820447036,5173304

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

%C A003239(n) is the number of 2n-bead balanced binary necklaces. A022553(n) among them are primitive.

%C The remaining a(n) necklaces are periodic.

%C Sequences counting 2n-bead balanced binary necklaces:

%C                        primitive  imprimitive

%C                      +-----------------------+---------+

%C   self-complementary |  A000048     A115118  | A000013 |

%C    complement pairs  |  A383904     A387130  | A386388 |

%C                      +-----------------------+---------+

%C                      |  A022553      this    | A003239 |

%C                      +-----------------------+---------+

%H Tilman Piesk, <a href="/A386946/b386946.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%e   n | A003239(n) A022553(n) | a(n)

%e   0 |         1          1  |   0

%e   1 |         1          1  |   0

%e   2 |         2          1  |   1

%e   3 |         4          3  |   1

%e   4 |        10          8  |   2

%e   5 |        26         25  |   1

%e   6 |        80         75  |   5

%e   7 |       246        245  |   1

%e   8 |       810        800  |  10

%e   9 |      2704       2700  |   4

%e  10 |      9252       9225  |  27

%e  11 |     32066      32065  |   1

%e  12 |    112720     112632  |  88

%e  13 |    400024     400023  |   1

%e  14 |   1432860    1432613  | 247

%e  15 |   5170604    5170575  |  29

%e  16 |  18784170   18783360  | 810

%e 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.

%K nonn

%O 0,5

%A _Tilman Piesk_, Aug 10 2025