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A353696
Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.
2
0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192
OFFSET
1,3
COMMENTS
First differs from the non-consecutive version A353431 in lacking 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are a subsequence but not a consecutive subsequence.
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.
EXAMPLE
The terms together with their corresponding compositions begin:
0: ()
1: (1)
2: (2)
4: (3)
8: (4)
10: (2,2)
16: (5)
32: (6)
43: (2,2,1,1)
58: (1,1,2,2)
64: (7)
128: (8)
256: (9)
292: (3,3,3)
349: (2,2,1,1,2,1)
442: (1,2,1,1,2,2)
512: (10)
586: (3,3,2,2)
676: (2,2,3,3)
697: (2,2,1,1,3,1)
826: (1,3,1,1,2,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y, {i, j}], {i, Length[y]}, {j, i, Length[y]}], Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Select[Range[0, 1000], yoyQ[stc[#]]&]
CROSSREFS
Non-recursive non-consecutive for partitions: A325755, counted by A325702.
Non-consecutive: A353431, counted by A353391.
Non-consecutive for partitions: A353393, counted by A353426.
Non-recursive non-consecutive: A353402, counted by A353390.
Counted by: A353430.
Non-recursive: A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, run-lengths A333769.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, multisets A225620, sets A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Sequence in context: A083655 A335404 A353431 * A271816 A097210 A097214
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 22 2022
STATUS
approved