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 A385665 #83 Aug 08 2025 19:03:00 %S A385665 1,1,1,1,0,1,2,1,0,1,3,0,0,0,1,5,1,1,0,0,1,9,0,0,0,0,0,1,16,2,0,1,0,0, %T A385665 0,1,28,0,1,0,0,0,0,0,1,51,3,0,0,1,0,0,0,0,1,93,0,0,0,0,0,0,0,0,0,1, %U A385665 170,5,2,1,0,1,0,0,0,0,0,1,315,0,0,0,0,0,0,0,0,0,0,0,1 %N A385665 Triangle read by rows: T(n,k) is the number of 2n-bead balanced bicolor necklaces that can be rotated into their complements in k different ways. %C A385665 Let X = A003239, Y = A000013, Z = A000048. %C A385665 Rotations producing the complementary and the same necklace: CR and SR %C A385665 There are X(n) balanced bicolor necklaces (BBN) of length 2n. (Central numbers of A047996.) %C A385665 Y(n) among them are self-complementary (SCBBN). (They can be rotated so that all beads change color.) %C A385665 Z(n) among those are primitive (not periodic). Each has a unique CR and SR. (SR is trivial rotation.) %C A385665 The other Y(n)-Z(n) = A115118(n) SCBBN have multiple CR and SR. %C A385665 T(n,k) SCBBN have k different CR and SR. %C A385665 Column 1 is Z. The other columns have the same positive entries, each preceded by k-1 zeros. %C A385665 One could add a column 0 to this triangle, whose entries would be X(n)-Y(n) = 2*A386388(n). %C A385665 Triangle A385666 does the same for SR of all BBN. %H A385665 Tilman Piesk, <a href="/A385665/b385665.txt">Rows 1..32, flattened</a> %H A385665 Tilman Piesk, <a href="/A045654/a045654_4.txt">Triangle T(n,k)*2*n/k with row sums A045654(n)</a> %F A385665 T(n,k) = A000048(n/k) iff n divisible by k, otherwise 0. %e A385665 Triangle begins: %e A385665 k 1 2 3 4 5 6 7 8 9 10 11 12 12 14 15 16 A000013(n) %e A385665 n %e A385665 1 1 . . . . . . . . . . . . . . . 1 %e A385665 2 1 1 . . . . . . . . . . . . . . 2 %e A385665 3 1 . 1 . . . . . . . . . . . . . 2 %e A385665 4 2 1 . 1 . . . . . . . . . . . . 4 %e A385665 5 3 . . . 1 . . . . . . . . . . . 4 %e A385665 6 5 1 1 . . 1 . . . . . . . . . . 8 %e A385665 7 9 . . . . . 1 . . . . . . . . . 10 %e A385665 8 16 2 . 1 . . . 1 . . . . . . . . 20 %e A385665 9 28 . 1 . . . . . 1 . . . . . . . 30 %e A385665 10 51 3 . . 1 . . . . 1 . . . . . . 56 %e A385665 11 93 . . . . . . . . . 1 . . . . . 94 %e A385665 12 170 5 2 1 . 1 . . . . . 1 . . . . 180 %e A385665 13 315 . . . . . . . . . . . 1 . . . 316 %e A385665 14 585 9 . . . . 1 . . . . . . 1 . . 596 %e A385665 15 1091 . 3 . 1 . . . . . . . . . 1 . 1096 %e A385665 16 2048 16 . 2 . . . 1 . . . . . . . 1 2068 %e A385665 Examples for n=4 with necklaces of length 8: %e A385665 T(4, 1) = 2 necklaces can be rotated into their complements in k=1 way: %e A385665 00001111 can be turned into 11110000 by rotating 4 places to the right. %e A385665 00101101 can be turned into 11010010 by rotating 4 places to the right. %e A385665 T(4, 2) = 1 necklace can be rotated into its complement in k=2 ways: %e A385665 00110011 can be turned into 11001100 by rotating 2 or 6 places to the right. %e A385665 T(4, 4) = 1 necklace can be rotated into its complement in k=4 ways: %e A385665 01010101 can be turned into 10101010 by rotating 1, 3, 5 or 7 places to the right. %Y A385665 Cf. A003239, A000013, A000048, A385666, A386388, A045654, A115118. %K A385665 nonn,tabl %O A385665 1,7 %A A385665 _Tilman Piesk_, Jul 06 2025