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A334968
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Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).
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19
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1, 2, 2, 3, 2, 4, 4, 4, 2, 4, 3, 5, 4, 5, 5, 5, 2, 4, 4, 6, 4, 6, 6, 6, 4, 6, 6, 6, 6, 6, 6, 6, 2, 4, 4, 6, 3, 7, 7, 7, 4, 7, 4, 7, 7, 7, 7, 7, 4, 6, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 7, 7, 2, 4, 4, 6, 4, 8, 8, 8, 4, 6, 6, 8, 6, 8, 8, 8, 4, 8, 6, 8, 6, 8, 8
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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The 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
Triangle begins:
1
2
2 3
2 4 4 4
2 4 3 5 4 5 5 5
2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Total/@Subsets[stc[n]]]], {n, 0, 100}]
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CROSSREFS
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Dominated by A124771 (number of contiguous subsequences).
Dominates A333257 (the contiguous case).
Dominated by A334299 (number of subsequences).
Positive subset-sums of partitions are counted by A276024 and A299701.
Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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