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 A334273 #6 Apr 28 2020 00:14:43
%S A334273 0,1,2,3,4,5,7,8,9,10,11,15,16,17,18,19,21,23,31,32,33,34,35,36,37,39,
%T A334273 42,43,45,47,63,64,65,66,67,68,69,71,73,74,75,77,79,85,87,91,95,127,
%U A334273 128,129,130,131,132,133,135,136,137,138,139,141,143,146,147
%N A334273 Numbers k such that the k-th composition in standard order is both a reversed necklace and a co-necklace.
%C A334273 A necklace is a finite sequence of positive integers that is lexicographically less than or equal to any cyclic rotation. Co-necklaces are defined similarly, except with greater instead of less.
%C A334273 The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
%e A334273 The sequence of all reversed necklace co-necklaces begins:
%e A334273 0: () 31: (1,1,1,1,1) 69: (4,2,1)
%e A334273 1: (1) 32: (6) 71: (4,1,1,1)
%e A334273 2: (2) 33: (5,1) 73: (3,3,1)
%e A334273 3: (1,1) 34: (4,2) 74: (3,2,2)
%e A334273 4: (3) 35: (4,1,1) 75: (3,2,1,1)
%e A334273 5: (2,1) 36: (3,3) 77: (3,1,2,1)
%e A334273 7: (1,1,1) 37: (3,2,1) 79: (3,1,1,1,1)
%e A334273 8: (4) 39: (3,1,1,1) 85: (2,2,2,1)
%e A334273 9: (3,1) 42: (2,2,2) 87: (2,2,1,1,1)
%e A334273 10: (2,2) 43: (2,2,1,1) 91: (2,1,2,1,1)
%e A334273 11: (2,1,1) 45: (2,1,2,1) 95: (2,1,1,1,1,1)
%e A334273 15: (1,1,1,1) 47: (2,1,1,1,1) 127: (1,1,1,1,1,1,1)
%e A334273 16: (5) 63: (1,1,1,1,1,1) 128: (8)
%e A334273 17: (4,1) 64: (7) 129: (7,1)
%e A334273 18: (3,2) 65: (6,1) 130: (6,2)
%e A334273 19: (3,1,1) 66: (5,2) 131: (6,1,1)
%e A334273 21: (2,2,1) 67: (5,1,1) 132: (5,3)
%e A334273 23: (2,1,1,1) 68: (4,3) 133: (5,2,1)
%t A334273 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334273 neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A334273 coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
%t A334273 Select[Range[0,100],neckQ[Reverse[stc[#]]]&&coneckQ[stc[#]]&]
%Y A334273 The aperiodic case is A334266.
%Y A334273 Compositions of this type are counted by A334271.
%Y A334273 Normal sequences of this type are counted by A334272.
%Y A334273 Another ranking of the same compositions is A334274 (binary expansion).
%Y A334273 Binary (or reversed binary) necklaces are counted by A000031.
%Y A334273 Necklace compositions are counted by A008965.
%Y A334273 All of the following pertain to compositions in standard order (A066099):
%Y A334273 - Necklaces are A065609.
%Y A334273 - Reversed necklaces are A333943.
%Y A334273 - Co-necklaces are A333764.
%Y A334273 - Reversed co-necklaces are A328595.
%Y A334273 - Lyndon words are A275692.
%Y A334273 - Co-Lyndon words are A326774.
%Y A334273 - Reversed Lyndon words are A334265.
%Y A334273 - Reversed co-Lyndon words are A328596.
%Y A334273 - Aperiodic compositions are A328594.
%Y A334273 Cf. A019536, A034691, A059966, A060223, A329138, A334269, A334270.
%K A334273 nonn
%O A334273 1,3
%A A334273 _Gus Wiseman_, Apr 25 2020