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A353848
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Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.
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33
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0, 1, 2, 3, 4, 7, 8, 10, 11, 14, 15, 16, 31, 32, 36, 39, 42, 46, 59, 60, 63, 64, 127, 128, 136, 138, 143, 168, 170, 175, 187, 238, 248, 250, 255, 256, 292, 316, 487, 511, 512, 528, 543, 682, 750, 955, 1008, 1023, 1024, 2047, 2048, 2080, 2084, 2090, 2111, 2184
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OFFSET
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0,3
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COMMENTS
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Every sequence can be uniquely split into non-overlapping runs, read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).
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 terms together with their binary expansions and corresponding compositions begin:
0: 0 ()
1: 1 (1)
2: 10 (2)
3: 11 (1,1)
4: 100 (3)
7: 111 (1,1,1)
8: 1000 (4)
10: 1010 (2,2)
11: 1011 (2,1,1)
14: 1110 (1,1,2)
15: 1111 (1,1,1,1)
16: 10000 (5)
31: 11111 (1,1,1,1,1)
32: 100000 (6)
36: 100100 (3,3)
39: 100111 (3,1,1,1)
42: 101010 (2,2,2)
46: 101110 (2,1,1,2)
59: 111011 (1,1,2,1,1)
60: 111100 (1,1,1,3)
For example:
- The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
- The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
- The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], SameQ@@Total/@Split[stc[#]]&]
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CROSSREFS
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Standard compositions are listed by A066099.
For equal lengths instead of sums we have A353744, counted by A329738.
These compositions are counted by A353851.
The run-sums themselves are listed by A353932, with A353849 distinct terms.
A005811 counts runs in binary expansion.
A351014 counts distinct runs in standard compositions, firsts A351015.
A353847 represents the run-sum transformation for compositions.
A353860 counts collapsible compositions.
A353863 counts run-sum-complete partitions.
Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.
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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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