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

%I A386946 #15 Aug 27 2025 18:20:26
%S A386946 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 A386946 113520,26,400025,2704,1432868,1,5179905,1,18784170,32069,68635479,
%U A386946 271,252201136,1,930138523,400027,3446168660,1,12817096533,1,47820447036,5173304
%N A386946 a(n) is the number of imprimitive (periodic) 2n-bead balanced binary necklaces.
%C A386946 A003239(n) is the number of 2n-bead balanced binary necklaces. A022553(n) among them are primitive.
%C A386946 The remaining a(n) necklaces are periodic.
%C A386946 Sequences counting 2n-bead balanced binary necklaces:
%C A386946                        primitive  imprimitive
%C A386946                      +-----------------------+---------+
%C A386946   self-complementary |  A000048     A115118  | A000013 |
%C A386946    complement pairs  |  A383904     A387130  | A386388 |
%C A386946                      +-----------------------+---------+
%C A386946                      |  A022553      this    | A003239 |
%C A386946                      +-----------------------+---------+
%H A386946 Tilman Piesk, <a href="/A386946/b386946.txt">Table of n, a(n) for n = 0..1000</a>
%F A386946 a(n) = A003239(n) - A022553(n).
%F A386946 a(n) = A115118(n) + 2 * A387130(n).
%e A386946   n | A003239(n) A022553(n) | a(n)
%e A386946   0 |         1          1  |   0
%e A386946   1 |         1          1  |   0
%e A386946   2 |         2          1  |   1
%e A386946   3 |         4          3  |   1
%e A386946   4 |        10          8  |   2
%e A386946   5 |        26         25  |   1
%e A386946   6 |        80         75  |   5
%e A386946   7 |       246        245  |   1
%e A386946   8 |       810        800  |  10
%e A386946   9 |      2704       2700  |   4
%e A386946  10 |      9252       9225  |  27
%e A386946  11 |     32066      32065  |   1
%e A386946  12 |    112720     112632  |  88
%e A386946  13 |    400024     400023  |   1
%e A386946  14 |   1432860    1432613  | 247
%e A386946  15 |   5170604    5170575  |  29
%e A386946  16 |  18784170   18783360  | 810
%e A386946 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 A386946 nonn,new
%O A386946 0,5
%A A386946 _Tilman Piesk_, Aug 10 2025