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 A385666 #19 Aug 27 2025 22:03:23 %S A385666 1,1,1,3,0,1,8,1,0,1,25,0,0,0,1,75,3,1,0,0,1,245,0,0,0,0,0,1,800,8,0, %T A385666 1,0,0,0,1,2700,0,3,0,0,0,0,0,1,9225,25,0,0,1,0,0,0,0,1,32065,0,0,0,0, %U A385666 0,0,0,0,0,1,112632,75,8,3,0,1,0,0,0,0,0,1,400023 %N A385666 Triangle read by rows: T(n,k) is the number of 2n-bead balanced binary necklaces with period length 2n/k. %C A385666 There are A003239(n) balanced binary necklaces of length 2n. (Central numbers of A047996.) %C A385666 T(n,k) is the number of those that can be rotated into themselves in k different ways (at least 1 for the trivial rotation). %C A385666 A022553(n) necklaces (corresponding to Lyndon words) have only the trivial rotation. %C A385666 All columns have the same positive entries, each preceded by k-1 zeros. %C A385666 Compare triangle A385665, which counts only self-complementary balanced binary necklaces. %H A385666 Tilman Piesk, <a href="/A385666/b385666.txt">Rows 1..32, flattened</a> %H A385666 Tilman Piesk, <a href="/A385666/a385666.txt">Triangle T(n,k)*2*n/k with row sums 2n choose n</a> %H A385666 Tilman Piesk, <a href="/A385666/a385666_2.txt">List of imprimitive necklaces for n=1...15</a> %H A385666 Tilman Piesk, <a href="/A385666/a385666_3.txt">List of primitive necklaces for n=1...8</a> %F A385666 T(n,k) = A022553(n/k) iff n divisible by k, otherwise 0. %e A385666 Triangle begins: %e A385666 k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 A003239(n) %e A385666 n %e A385666 1 1 . . . . . . . . . . . . . . . 1 %e A385666 2 1 1 . . . . . . . . . . . . . . 2 %e A385666 3 3 . 1 . . . . . . . . . . . . . 4 %e A385666 4 8 1 . 1 . . . . . . . . . . . . 10 %e A385666 5 25 . . . 1 . . . . . . . . . . . 26 %e A385666 6 75 3 1 . . 1 . . . . . . . . . . 80 %e A385666 7 245 . . . . . 1 . . . . . . . . . 246 %e A385666 8 800 8 . 1 . . . 1 . . . . . . . . 810 %e A385666 9 2700 . 3 . . . . . 1 . . . . . . . 2704 %e A385666 10 9225 25 . . 1 . . . . 1 . . . . . . 9252 %e A385666 11 32065 . . . . . . . . . 1 . . . . . 32066 %e A385666 12 112632 75 8 3 . 1 . . . . . 1 . . . . 112720 %e A385666 13 400023 . . . . . . . . . . . 1 . . . 400024 %e A385666 14 1432613 245 . . . . 1 . . . . . . 1 . . 1432860 %e A385666 15 5170575 . 25 . 3 . . . . . . . . . 1 . 5170604 %e A385666 16 18783360 800 . 8 . . . 1 . . . . . . . 1 18784170 %e A385666 Examples for n=4 with necklaces of length 8: %e A385666 T(4, 1) = 8 necklaces have k=1 rotation, i.e. rotating 0 places: %e A385666 00001111, 00010111, 00011011, 00011101, 00100111, 00101011, 00101101, 00110101 %e A385666 T(4, 2) = 1 necklace has k=2 rotations: %e A385666 00110011 can be rotated onto itself by rotating 0 or 4 places. %e A385666 T(4, 4) = 1 necklace has k=4 rotations: %e A385666 01010101 can be rotated onto itself by rotating 0, 2, 4 or 6 places. %Y A385666 Cf. A022553, A003239, A385665. %K A385666 nonn,tabl,changed %O A385666 1,4 %A A385666 _Tilman Piesk_, Jul 16 2025