A333632 - OEIS

%I #10 Apr 28 2020 18:37:26

%S 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 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 2,3,3,4,2,3,3,4,3,4,4,5,2,3,3,4,3,4,4

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

%C 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 a(n) = A000120(n)/A138904(n) = A302291(n) - A023416(n)/A138904(n).

%e The a(299) = 5 rotations:

%e   (1,1,3,2,2)

%e   (1,3,2,2,1)

%e   (3,2,2,1,1)

%e   (2,2,1,1,3)

%e   (2,1,1,3,2)

%e The a(9933) = 4 rotations:

%e   (1,2,1,3,1,2,1,3)

%e   (1,3,1,2,1,3,1,2)

%e   (2,1,3,1,2,1,3,1)

%e   (3,1,2,1,3,1,2,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[Length[Union[Array[RotateRight[stc[n],#]&,DigitCount[n,2,1]]]],{n,0,100}]

%Y Aperiodic compositions are counted by A000740.

%Y Aperiodic binary words are counted by A027375.

%Y The orderless period of prime indices is A052409.

%Y Numbers whose binary expansion is periodic are A121016.

%Y Periodic compositions are counted by A178472.

%Y The version for binary expansion is A302291.

%Y Numbers whose prime signature is aperiodic are A329139.

%Y Compositions by number of distinct rotations are A333941.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Necklaces are A065609.

%Y - Sum is A070939.

%Y - Equal runs are counted by A124767.

%Y - Rotational symmetries are counted by A138904.

%Y - Strict compositions are A233564.

%Y - Constant compositions are A272919.

%Y - Lyndon compositions are A275692.

%Y - Co-Lyndon compositions are A326774.

%Y - Aperiodic compositions are A328594.

%Y - Rotational period is A333632 (this sequence).

%Y - Co-necklaces are A333764.

%Y - Reversed necklaces are A333943.

%Y Cf. A000031, A001037, A008965, A019536, A211100, A328595, A328596, A329312, A329313, A329326.

%K nonn

%O 0,6

%A _Gus Wiseman_, Apr 12 2020