proposed

approved

editing

proposed

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

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

a(n) is the row number in A066099 of the odd bisection of the n-th row of A066099. 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.

Length of binary expansion of a(n) is A000120(n)/2 rounded up.

Length of binary expansion Sum of a(n) -th standard composition is A000120A209281(n+1)/2 rounded up.

Sum of binary expansion of a(n) is A209281(n).

Row a(n) is the row number in A066099 of the odd bisection of the n-th composition in standard orderrow of A066099. 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.

Length of binary expansion of a(n) is A209281A000120(n)/2 rounded up.

[check]Positions Sum of binary expansion of first appearances are A290259a(n) is A209281(n).

Positions of first appearances are A290259.

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

Cf. A000009, A000070, A000097, A000302, A000346, `A001792, A008549, A025047, A088218, `A124754, A131577, A294175, A344606, A344653, `A346633, A346697 (reverse: A346699).

allocated for Gus WisemanThe 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

0,3

Row number in A066099 of the odd bisection of the n-th composition in standard order. 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.

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

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.

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}]

Positions of 1's are A003945.

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

Length of binary expansion of a(n) is A209281(n).

[check]Positions of first appearances are A290259.

The version for prime indices is A346703.

The even-indexed version is A346705.

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, A000070, A000097, A000302, A000346, `A001792, A008549, A025047, A088218, `A124754, A131577, A294175, A344606, A344653, `A346633, A346697 (reverse: A346699).

allocated

nonn

Gus Wiseman, Aug 12 2021

approved

editing

allocated for Gus Wiseman

allocated

approved