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A357134
Take the k-th composition in standard order for each part k of the n-th composition in standard order; then set a(n) to be the index (in standard order) of the concatenation.
9
0, 1, 2, 3, 3, 5, 6, 7, 4, 7, 10, 11, 7, 13, 14, 15, 5, 9, 14, 15, 11, 21, 22, 23, 12, 15, 26, 27, 15, 29, 30, 31, 6, 11, 18, 19, 15, 29, 30, 31, 20, 23, 42, 43, 23, 45, 46, 47, 13, 25, 30, 31, 27, 53, 54, 55, 28, 31, 58, 59, 31, 61, 62, 63, 7, 13, 22, 23, 19
OFFSET
0,3
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
For n > 0, the value n appears A048896(n - 1) times.
Row a(n) of A066099 = row n of A357135.
EXAMPLE
The terms together with their corresponding standard compositions begin:
0: ()
1: (1)
2: (2)
3: (1,1)
3: (1,1)
5: (2,1)
6: (1,2)
7: (1,1,1)
4: (3)
7: (1,1,1)
10: (2,2)
11: (2,1,1)
7: (1,1,1)
13: (1,2,1)
14: (1,1,2)
15: (1,1,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[stcinv[Join@@stc/@stc[n]], {n, 0, 30}]
CROSSREFS
See link for sequences related to standard compositions.
The version for Heinz numbers of partitions is A003963.
The vertex-degrees are A048896.
The a(n)-th composition in standard order is row n of A357135.
Sequence in context: A212010 A366418 A328745 * A003967 A349390 A099209
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 24 2022
STATUS
approved