cp's OEIS Frontend

A346633 Sum of even-indexed parts (even bisection) of the n-th composition in standard order.

%I A346633 #8 Aug 02 2021 07:56:30
%S A346633 0,0,0,1,0,1,2,1,0,1,2,1,3,2,1,2,0,1,2,1,3,2,1,2,4,3,2,3,1,2,3,2,0,1,
%T A346633 2,1,3,2,1,2,4,3,2,3,1,2,3,2,5,4,3,4,2,3,4,3,1,2,3,2,4,3,2,3,0,1,2,1,
%U A346633 3,2,1,2,4,3,2,3,1,2,3,2,5,4,3,4,2,3,4
%N A346633 Sum of even-indexed parts (even bisection) of the n-th composition in standard order.
%C A346633 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. This gives a bijective correspondence between nonnegative integers and integer compositions.
%F A346633 a(n) = (A029837(n) - A124754(n))/2.
%F A346633 a(n) = A029837(n) - A209281(n + 1).
%F A346633 a(n) = A124754(n) + A209281(n + 1).
%e A346633 Composition number 741 in standard order is (2,1,1,3,2,1), so a(741) = 1 + 3 + 1 = 5.
%t A346633 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A346633 Table[Total[Last/@Partition[Append[stc[n],0],2]],{n,0,100}]
%Y A346633 Including odd-indexed parts gives A029837.
%Y A346633 Subtracting from the odd version gives A124754.
%Y A346633 Positions of zeros are A131577.
%Y A346633 The odd-indexed version is A209281(n+1).
%Y A346633 The version for prime indices is A346698 (reverse: A346700).
%Y A346633 A000120 and A080791 count binary digits 1 and 0, with difference A145037.
%Y A346633 A011782 counts compositions.
%Y A346633 A056239 adds up prime indices, row sums of A112798.
%Y A346633 A097805 counts compositions by alternating (or reverse-alternating) sum.
%Y A346633 A316524 gives the alternating sum of prime indices (reverse: A344616).
%Y A346633 A325534 counts separable partitions, ranked by A335433.
%Y A346633 A325535 counts inseparable partitions, ranked by A335448.
%Y A346633 A345197 counts compositions by sum, length, and alternating sum.
%Y A346633 Cf. A000070, A000097, A008549, A025047, A088218, A344606, A344653, A346697.
%K A346633 nonn
%O A346633 0,7
%A A346633 _Gus Wiseman_, Aug 01 2021