A346705 - OEIS

%I #9 Aug 23 2021 10:24:55

%S 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,

%T 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,

%U 1,4,2,1,3,8,4,2,5,1,3,6,3,16,8,4,9,2,5

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

%C 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.

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

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

%e 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.

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

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

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

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

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

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

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

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

%Y An unordered version is A346704, sums A346698.

%Y A011782 counts compositions.

%Y A029837 gives length of binary expansion, or sometimes A070939.

%Y A066099 lists compositions in standard order.

%Y A097805 counts compositions by alternating sum.

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

%K nonn

%O 0,7

%A _Gus Wiseman_, Aug 19 2021