cp's OEIS Frontend

A346702 The a(n)-th composition in standard order is the odd bisection of the n-th composition in standard order.

%I A346702 #8 Aug 20 2021 00:24:50
%S A346702 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,
%T A346702 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,
%U A346702 3,7,14,7,64,32,16,33,8,17,34,17,4,9,18,9,36,18
%N A346702 The a(n)-th composition in standard order is the odd bisection of the n-th composition in standard order.
%C A346702 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 A346702 a(n) is the row number in A066099 of the odd bisection of the n-th row of A066099.
%F A346702 A029837(a(n)) = A209281(n).
%e A346702 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.
%t A346702 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}]
%Y A346702 Length of the a(n)-th standard composition is A000120(n)/2 rounded up.
%Y A346702 Positions of 1's are A003945.
%Y A346702 Positions of 2's (and zero) are A083575.
%Y A346702 Sum of the a(n)-th standard composition is A209281(n+1).
%Y A346702 Positions of first appearances are A290259.
%Y A346702 The version for prime indices is A346703.
%Y A346702 The version for even bisection is A346705, with sums A346633.
%Y A346702 A000120 and A080791 count binary digits 1 and 0, with difference A145037.
%Y A346702 A011782 counts compositions.
%Y A346702 A029837 gives length of binary expansion.
%Y A346702 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A346702 A345197 counts compositions by sum, length, and alternating sum.
%Y A346702 Cf. A000009, A000302, A000346, A025047, A088218, A124754, A294175, A346697 (reverse: A346699).
%K A346702 nonn
%O A346702 0,3
%A A346702 _Gus Wiseman_, Aug 12 2021