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A357134
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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.
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9
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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
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OFFSET
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0,3
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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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For n > 0, the value n appears A048896(n - 1) times.
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EXAMPLE
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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)
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MATHEMATICA
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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}]
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CROSSREFS
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See link for sequences related to standard compositions.
The version for Heinz numbers of partitions is A003963.
The a(n)-th composition in standard order is row n of A357135.
Cf. A000120, A001511, A029931, A058891, A070939, A096111, A329395, A333766, A335404, A357137, A357180.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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