This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A387130 #29 Aug 27 2025 18:20:38 %S A387130 0,0,0,0,0,0,1,0,3,1,11,0,39,0,118,12,395,0,1372,0,4601,119,15986,0, %T A387130 56662,11,199854,1337,716135,0,2589376,0,9391051,15987,34315811,129, %U A387130 126096824,0,465062362,199855,1723071186,0,6408523001,0,23910175807,2586090,89493721076 %N A387130 a(n) is the number of complement pairs of imprimitive (periodic) 2n-bead balanced binary necklaces. %C A387130 A386946(n) is the number of primitive 2n-bead balanced binary necklaces (corresponding to Lyndon words), and A115118 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 A387130 Doubled entries: 0, 0, 0, 0, 0, 0, 2, 0, 6, 2, 22, 0, 78, 0, 236, 24, 790, 0, 2744, ... %C A387130 Sequences counting 2n-bead balanced binary necklaces: %C A387130 primitive imprimitive %C A387130 +-----------------------+---------+ %C A387130 self-complementary | A000048 A115118 | A000013 | %C A387130 complement pairs | A383904 this | A386388 | %C A387130 +-----------------------+---------+ %C A387130 | A022553 A386946 | A003239 | %C A387130 +-----------------------+---------+ %H A387130 Tilman Piesk, <a href="/A387130/b387130.txt">Table of n, a(n) for n = 0..1000</a> %F A387130 a(n) = (A386946(n) - A115118(n)) / 2. %F A387130 a(n) = A386388(n) - A383904(n). %e A387130 n | A386946(n) A115118(n) | 2*a(n) a(n) | A386388(n) A383904(n) %e A387130 0 | 0 0 | 0 0 | 0 0 %e A387130 1 | 0 0 | 0 0 | 0 0 %e A387130 2 | 1 1 | 0 0 | 0 0 %e A387130 3 | 1 1 | 0 0 | 1 1 %e A387130 4 | 2 2 | 0 0 | 3 3 %e A387130 5 | 1 1 | 0 0 | 11 11 %e A387130 6 | 5 3 | 2 1 | 36 35 %e A387130 7 | 1 1 | 0 0 | 118 118 %e A387130 8 | 10 4 | 6 3 | 395 392 %e A387130 9 | 4 2 | 2 1 | 1337 1336 %e A387130 10 | 27 5 | 22 11 | 4598 4587 %e A387130 11 | 1 1 | 0 0 | 15986 15986 %e A387130 12 | 88 10 | 78 39 | 56270 56231 %e A387130 13 | 1 1 | 0 0 | 199854 199854 %e A387130 14 | 247 11 | 236 118 | 716132 716014 %e A387130 15 | 29 5 | 24 12 | 2584754 2584742 %e A387130 16 | 810 20 | 790 395 | 9391051 9390656 %e A387130 Examples for n=8 with necklaces of length 16: %e A387130 The total number of necklaces is A003239(8) = 810. %e A387130 A022553(8) = 800 of them are primitive. %e A387130 The other A386946(8) = 10 are periodic. %e A387130 A115118(8) = 4 among those are self-complementary: %e A387130 0000111100001111 %e A387130 0010110100101101 %e A387130 0011001100110011 %e A387130 0101010101010101 %e A387130 The remaining 6 necklaces form a(8) = 3 complement pairs: %e A387130 0001011100010111 0001110100011101 %e A387130 0001101100011011 0010011100100111 %e A387130 0010101100101011 0011010100110101 %Y A387130 Cf. A386946, A115118, A386388, A383904. %K A387130 nonn,new %O A387130 0,9 %A A387130 _Tilman Piesk_, Aug 17 2025