A346702 - OEIS

A346702

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

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, 32, 16, 8, 17, 4, 9, 18, 9, 2, 5, 10, 5, 20, 10, 5, 11, 1, 3, 6, 3, 12, 6, 3, 7, 24, 12, 6, 13, 3, 7, 14, 7, 64, 32, 16, 33, 8, 17, 34, 17, 4, 9, 18, 9, 36, 18

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OFFSET

0,3

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 odd bisection of the n-th row of A066099.

LINKS

Table of n, a(n) for n=0..77.

FORMULA

A029837(a(n)) = A209281(n).

EXAMPLE

Composition number 741 in standard order is (2,1,1,3,2,1), with odd bisection (2,1,2), which is composition number 22 in standard order, hence a(741) = 22.

MATHEMATICA

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

CROSSREFS

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

Positions of 1's are A003945.

Positions of 2's (and zero) are A083575.

Sum of the a(n)-th standard composition is A209281(n+1).

Positions of first appearances are A290259.

The version for prime indices is A346703.

The version for even bisection is A346705, with sums A346633.

A000120 and A080791 count binary digits 1 and 0, with difference A145037.

A011782 counts compositions.

A029837 gives length of binary expansion.

A097805 counts compositions by alternating (or reverse-alternating) sum.

A345197 counts compositions by sum, length, and alternating sum.

Cf. A000009, A000302, A000346, A025047, A088218, A124754, A294175, A346697 (reverse: A346699).

Sequence in context: A286321 A349192 A118235 * A304624 A083653 A278290

Adjacent sequences: A346699 A346700 A346701 * A346703 A346704 A346705

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 12 2021

STATUS

approved