A346705 The a(n)-th composition in standard order is the even bisection of the n-th composition in standard order.

0, 0, 0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 4, 2, 1, 3, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5, 10, 5, 1, 3, 6, 3, 12, 6, 3, 7, 0, 1, 2, 1, 4, 2, 1, 3, 8, 4, 2, 5, 1, 3, 6, 3, 16, 8, 4, 9, 2, 5

Offset: 0

Author

Gus Wiseman, Aug 19 2021

Keywords

• nonn

Comments

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.

a(n) is the row number in A066099 of the even bisection (even-indexed terms) of the n-th row of A066099.

Examples

			Composition number 741 in standard order is (2,1,1,3,2,1), with even bisection (1,3,1), which is composition number 25 in standard order, so a(741) = 25.
		

Crossrefs

Length of the a(n)-th standard composition is A000120(n)/2 rounded down.

Positions of first appearances appear to be A088698, sorted: A277335.

The version for reversed prime indices appears to be A329888, sums A346700.

Sum of the a(n)-th standard composition is A346633.

An unordered reverse version for odd bisection is A346701, sums A346699.

The version for odd bisection is A346702, sums A209281(n+1).

An unordered version for odd bisection is A346703, sums A346697.

An unordered version is A346704, sums A346698.

A011782 counts compositions.

A029837 gives length of binary expansion, or sometimes A070939.

A066099 lists compositions in standard order.

A097805 counts compositions by alternating sum.

Cf. A000302, A025047, A088218, A124754, A124767, A228351, A290258, A290259, A344618, A345197.

Programs

Mathematica

Table[Total[2^Accumulate[Reverse[Last/@Partition[ Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse,2]]]]/2,{n,0,100}]

Formula

A029837(a(n)) = A346633(n).