cp's OEIS Frontend

A334274 Numbers k such that the k-th composition in standard order is both a necklace and a reversed co-necklace.

%I A334274 #7 Apr 28 2020 00:14:50
%S A334274 0,1,2,3,4,6,7,8,10,12,14,15,16,20,24,26,28,30,31,32,36,40,42,48,52,
%T A334274 54,56,58,60,62,63,64,72,80,84,96,100,104,106,108,112,116,118,120,122,
%U A334274 124,126,127,128,136,144,160,164,168,170,192,200,204,208,212,216
%N A334274 Numbers k such that the k-th composition in standard order is both a necklace and a reversed co-necklace.
%C A334274 Also numbers whose binary expansion is both a reversed necklace and a co-necklace.
%C A334274 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 A334274 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 A334274 The sequence of all reversed co-necklace necklaces begins:
%e A334274     0: ()            31: (1,1,1,1,1)      100: (1,3,3)
%e A334274     1: (1)           32: (6)              104: (1,2,4)
%e A334274     2: (2)           36: (3,3)            106: (1,2,2,2)
%e A334274     3: (1,1)         40: (2,4)            108: (1,2,1,3)
%e A334274     4: (3)           42: (2,2,2)          112: (1,1,5)
%e A334274     6: (1,2)         48: (1,5)            116: (1,1,2,3)
%e A334274     7: (1,1,1)       52: (1,2,3)          118: (1,1,2,1,2)
%e A334274     8: (4)           54: (1,2,1,2)        120: (1,1,1,4)
%e A334274    10: (2,2)         56: (1,1,4)          122: (1,1,1,2,2)
%e A334274    12: (1,3)         58: (1,1,2,2)        124: (1,1,1,1,3)
%e A334274    14: (1,1,2)       60: (1,1,1,3)        126: (1,1,1,1,1,2)
%e A334274    15: (1,1,1,1)     62: (1,1,1,1,2)      127: (1,1,1,1,1,1,1)
%e A334274    16: (5)           63: (1,1,1,1,1,1)    128: (8)
%e A334274    20: (2,3)         64: (7)              136: (4,4)
%e A334274    24: (1,4)         72: (3,4)            144: (3,5)
%e A334274    26: (1,2,2)       80: (2,5)            160: (2,6)
%e A334274    28: (1,1,3)       84: (2,2,3)          164: (2,3,3)
%e A334274    30: (1,1,1,2)     96: (1,6)            168: (2,2,4)
%t A334274 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334274 neckQ[q_]:=Length[q]==0||Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
%t A334274 coneckQ[q_]:=Length[q]==0||Array[OrderedQ[{RotateRight[q,#],q}]&,Length[q]-1,1,And];
%t A334274 Select[Range[0,100],neckQ[stc[#]]&&coneckQ[Reverse[stc[#]]]&]
%Y A334274 The aperiodic case is A334267.
%Y A334274 Compositions of this type are counted by A334271.
%Y A334274 Normal sequences of this type are counted by A334272.
%Y A334274 Binary (or reversed binary) necklaces are counted by A000031.
%Y A334274 Necklace compositions are counted by A008965.
%Y A334274 All of the following pertain to compositions in standard order (A066099):
%Y A334274 - Necklaces are A065609.
%Y A334274 - Reversed necklaces are A333943.
%Y A334274 - Co-necklaces are A333764.
%Y A334274 - Reversed co-necklaces are A328595.
%Y A334274 - Lyndon words are A275692.
%Y A334274 - Co-Lyndon words are A326774.
%Y A334274 - Reversed Lyndon words are A334265.
%Y A334274 - Reversed co-Lyndon words are A328596.
%Y A334274 - Aperiodic compositions are A328594.
%Y A334274 Cf. A019536, A034691, A059966, A060223, A329138, A334266, A334269, A334270.
%K A334274 nonn
%O A334274 1,3
%A A334274 _Gus Wiseman_, Apr 25 2020