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A333632 Rotational period of the k-th composition in standard order; a(0) = 0.

%I A333632 #10 Apr 28 2020 18:37:26
%S A333632 0,1,1,1,1,2,2,1,1,2,1,3,2,3,3,1,1,2,2,3,2,3,3,4,2,3,3,4,3,4,4,1,1,2,
%T A333632 2,3,1,3,3,4,2,3,1,4,3,2,4,5,2,3,3,4,3,4,2,5,3,4,4,5,4,5,5,1,1,2,2,3,
%U A333632 2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4
%N A333632 Rotational period of the k-th composition in standard order; a(0) = 0.
%C A333632 A composition of n is a finite sequence of positive integers summing to n. 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.
%F A333632 a(n) = A000120(n)/A138904(n) = A302291(n) - A023416(n)/A138904(n).
%e A333632 The a(299) = 5 rotations:
%e A333632   (1,1,3,2,2)
%e A333632   (1,3,2,2,1)
%e A333632   (3,2,2,1,1)
%e A333632   (2,2,1,1,3)
%e A333632   (2,1,1,3,2)
%e A333632 The a(9933) = 4 rotations:
%e A333632   (1,2,1,3,1,2,1,3)
%e A333632   (1,3,1,2,1,3,1,2)
%e A333632   (2,1,3,1,2,1,3,1)
%e A333632   (3,1,2,1,3,1,2,1)
%t A333632 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333632 Table[Length[Union[Array[RotateRight[stc[n],#]&,DigitCount[n,2,1]]]],{n,0,100}]
%Y A333632 Aperiodic compositions are counted by A000740.
%Y A333632 Aperiodic binary words are counted by A027375.
%Y A333632 The orderless period of prime indices is A052409.
%Y A333632 Numbers whose binary expansion is periodic are A121016.
%Y A333632 Periodic compositions are counted by A178472.
%Y A333632 The version for binary expansion is A302291.
%Y A333632 Numbers whose prime signature is aperiodic are A329139.
%Y A333632 Compositions by number of distinct rotations are A333941.
%Y A333632 All of the following pertain to compositions in standard order (A066099):
%Y A333632 - Length is A000120.
%Y A333632 - Necklaces are A065609.
%Y A333632 - Sum is A070939.
%Y A333632 - Equal runs are counted by A124767.
%Y A333632 - Rotational symmetries are counted by A138904.
%Y A333632 - Strict compositions are A233564.
%Y A333632 - Constant compositions are A272919.
%Y A333632 - Lyndon compositions are A275692.
%Y A333632 - Co-Lyndon compositions are A326774.
%Y A333632 - Aperiodic compositions are A328594.
%Y A333632 - Rotational period is A333632 (this sequence).
%Y A333632 - Co-necklaces are A333764.
%Y A333632 - Reversed necklaces are A333943.
%Y A333632 Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.
%K A333632 nonn
%O A333632 0,6
%A A333632 _Gus Wiseman_, Apr 12 2020