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

%I A383904 #23 Aug 27 2025 18:20:19
%S A383904 0,0,0,1,3,11,35,118,392,1336,4587,15986,56231,199854,716014,2584742,
%T A383904 9390656,34315811,126039218,465062362,1723066193,6407806833,
%U A383904 23910159818,89493721076,335912335304,1264105728831,4768446686910,18027215660947,68291877609003
%N A383904 a(n) is the number of complement pairs of primitive 2n-bead balanced binary necklaces.
%C A383904 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.
%C A383904 Doubled entries: 0, 0, 0, 2, 6, 22, 70, 236, 784, 2672, 9174, 31972, 112462, 399708, 1432028, ...
%C A383904 Sequences counting 2n-bead balanced binary necklaces:
%C A383904                        primitive  imprimitive
%C A383904                      +-----------------------+---------+
%C A383904   self-complementary |  A000048     A115118  | A000013 |
%C A383904    complement pairs  |   this       A387130  | A386388 |
%C A383904                      +-----------------------+---------+
%C A383904                      |  A022553     A386946  | A003239 |
%C A383904                      +-----------------------+---------+
%H A383904 Tilman Piesk, <a href="/A383904/b383904.txt">Table of n, a(n) for n = 0..1000</a>
%F A383904 a(n) = (A022553(n) - A000048(n)) / 2.
%e A383904   n | A022553(n) A000048(n) | 2*a(n)    a(n)
%e A383904   0 |         1          1  |     0       0
%e A383904   1 |         1          1  |     0       0
%e A383904   2 |         1          1  |     0       0
%e A383904   3 |         3          1  |     2       1
%e A383904   4 |         8          2  |     6       3
%e A383904   5 |        25          3  |    22      11
%e A383904   6 |        75          5  |    70      35
%e A383904   7 |       245          9  |   236     118
%e A383904   8 |       800         16  |   784     392
%e A383904   9 |      2700         28  |  2672    1336
%e A383904  10 |      9225         51  |  9174    4587
%e A383904 Examples for n=5 with necklaces of length 10:
%e A383904 The total number of necklaces is A003239(5) = 26.
%e A383904 Only A386946(5) = 1 of them is periodic, namely 0101010101.
%e A383904 The other A022553(5) = 25 are primitive.
%e A383904 A000048(5) = 3 among those are self-complementary:
%e A383904  0000011111
%e A383904  0001011101
%e A383904  0010011011
%e A383904 The remaining 22 necklaces form a(5) = 11 complement pairs:
%e A383904  0000101111 0000111101
%e A383904  0000110111 0001111001
%e A383904  0000111011 0001001111
%e A383904  0001010111 0001110101
%e A383904  0001011011 0010011101
%e A383904  0001100111 0001110011
%e A383904  0001101011 0010100111
%e A383904  0001101101 0010010111
%e A383904  0010101011 0011010101
%e A383904  0010101101 0010110101
%e A383904  0010110011 0011001101
%K A383904 nonn,new
%O A383904 0,5
%A A383904 _Tilman Piesk_, Aug 07 2025