OFFSET
0,2
COMMENTS
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.
FORMULA
a(n) = A334300(n) + 1.
EXAMPLE
Triangle begins:
1
2
2 3
2 4 4 4
2 4 3 6 4 7 6 5
2 4 4 6 4 6 7 8 4 7 6 10 6 10 8 6
If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the subsequences of the composition with STC-number n:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 0 1 0 2 2 3 0 4 2 5 4 6 6 7
0 1 1 1 1 0 3 1 5 3 3
0 0 0 0 2 0 3 2 1
1 2 1 0
0 1 0
0
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Union[Subsets[stc[n]]]], {n, 0, 100}]
CROSSREFS
Row lengths are A011782.
Looking only at contiguous subsequences gives A124771.
Compositions where every subinterval has a different sum are A333222.
Knapsack compositions are A333223.
Contiguous positive subsequence-sums are counted by A333224.
Contiguous subsequence-sums are counted by A333257.
Disallowing empty subsequences gives A334300.
Subsequence-sums are counted by A334968.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 01 2020
STATUS
approved