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A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.

%I A333257 #5 Mar 21 2020 16:35:45
%S A333257 1,2,2,3,2,4,4,4,2,4,3,5,4,5,5,5,2,4,4,6,4,6,5,6,4,5,6,6,6,6,6,6,2,4,
%T A333257 4,6,3,6,6,7,4,7,4,7,6,7,6,7,4,5,7,7,6,7,7,7,6,7,7,7,7,7,7,7,2,4,4,6,
%U A333257 4,7,7,8,4,6,6,8,5,7,7,8,4,7,5,8,6,8,7
%N A333257 Number of distinct consecutive subsequence-sums of the k-th composition in standard order.
%C A333257 A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
%F A333257 a(n) = A333224(n) + 1.
%e A333257 The ninth composition in standard order is (3,1), which has consecutive subsequences (), (1), (3), (3,1), with sums 0, 1, 3, 4, so a(9) = 4.
%t A333257 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A333257 Table[Length[Union[ReplaceList[stc[n],{___,s___,___}:>Plus[s]]]],{n,0,100}]
%Y A333257 Dominated by A124771.
%Y A333257 Compositions where every subinterval has a different sum are counted by A169942 and A325677 and ranked by A333222, while the case of partitions is counted by A325768 and ranked by A325779.
%Y A333257 Positive subset-sums of partitions are counted by A276024 and A299701.
%Y A333257 Knapsack partitions are counted by A108917 and ranked by A299702.
%Y A333257 Knapsack compositions are counted by A325676 and A325687 and ranked by A333223.
%Y A333257 The version for Heinz numbers of partitions is A325770.
%Y A333257 Not allowing empty subsequences gives A333224.
%Y A333257 Cf. A000120, A029931, A048793, A059519, A066099, A070939, A114994, A124765, A124767, A233564, A272919, A325778, A333217.
%K A333257 nonn
%O A333257 0,2
%A A333257 _Gus Wiseman_, Mar 20 2020