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Positions of first appearances for n > 0 appear to be A088698, (sorted: A277335 or A290258).
The version for reversed prime indices is appears to be A329888, sums A346700.
The odd-indexed reverse version for reversed prime indices odd bisection is A346701, sums A346699.
The odd-indexed An unordered version for prime indices odd bisection is A346703, sums A346697.
The An unordered version for prime indices is A346704, sums A346698.
A000120 and A080791 count binary digits 1 and 0, difference A145037.
A029837 gives length of binary expansion (, or sometimes A070939).
A097805 counts compositions by alternating (or reverse-alternating) sum.
Cf. A000009, A000302, A000346, ~A003945, A025047, `A059893, ~A083575, A088218, A124754 (reverse: A344618), , A124767, `A228351, A290258, A290259, A294175, `A333219, A344618, A345197.
Table[Total[2^Accumulate[Reverse[Last/@Partition[ Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse, 2]]]]/2, {n, 0, 100}]
The version for reversed prime indices is A329888, sums A346700.
The odd-indexed version for odd bisection reversed prime indices is A346702, with A346701, sums A209281(n+1)A346699.
The odd-indexed version for prime indices odd bisection is A346703 (reverse: A346701, A346702, sums: A346699 A209281(n+1) with sums A346697.
The odd-indexed version for prime indices is A346704 (reverse: A329888, sums: A346700), with A346703, sums A346698A346697.
A000120 and A080791 count binary digits 1 and 0, with difference A145037The version for prime indices is A346704, sums A346698.
A000120 and A080791 count binary digits 1 and 0, difference A145037.
allocated for Gus WisemanThe 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
0,7
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.
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.
Table[Total[2^Accumulate[Reverse[Last/@Partition[Differences[Prepend[Join@@Position
Length of the a(n)-th standard composition is A000120(n)/2 rounded down.
Positions of first appearances for n > 0 appear to be A088698, (sorted: A277335 or A290258).
Sum of the a(n)-th standard composition is A346633.
The version for odd bisection is A346702, with sums A209281(n+1).
The odd-indexed version for prime indices is A346703 (reverse: A346701, sums: A346699) with sums A346697.
The version for prime indices is A346704 (reverse: A329888, sums: A346700), with sums A346698.
A000120 and A080791 count binary digits 1 and 0, with difference A145037.
A011782 counts compositions.
A029837 gives length of binary expansion (or sometimes A070939).
A066099 lists compositions in standard order.
A097805 counts compositions by alternating (or reverse-alternating) sum.
Cf. A000009, A000302, A000346, ~A003945, A025047, `A059893, ~A083575, A088218, A124754 (reverse: A344618), A124767, `A228351, A290259, A294175, `A333219, A345197.
allocated
nonn
Gus Wiseman, Aug 19 2021
approved
editing
allocated for Gus Wiseman
allocated
approved