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A373955
Numbers k such that the k-th integer composition in standard order contains two adjacent ones and no other runs.
1
3, 11, 14, 19, 27, 28, 29, 35, 46, 51, 56, 57, 67, 75, 78, 83, 91, 92, 93, 99, 110, 112, 113, 114, 116, 118, 131, 139, 142, 155, 156, 157, 163, 179, 184, 185, 195, 203, 206, 211, 219, 220, 221, 224, 225, 226, 229, 230, 232, 233, 236, 237, 259, 267, 270, 275
OFFSET
1,1
COMMENTS
Also numbers k such that the excess compression of the k-th integer composition in standard order is 1.
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.
postn of 1 in
EXAMPLE
The terms and corresponding compositions begin:
3: (1,1)
11: (2,1,1)
14: (1,1,2)
19: (3,1,1)
27: (1,2,1,1)
28: (1,1,3)
29: (1,1,2,1)
35: (4,1,1)
46: (2,1,1,2)
51: (1,3,1,1)
56: (1,1,4)
57: (1,1,3,1)
67: (5,1,1)
75: (3,2,1,1)
78: (3,1,1,2)
83: (2,3,1,1)
91: (2,1,2,1,1)
92: (2,1,1,3)
93: (2,1,1,2,1)
99: (1,4,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[100], Total[stc[#]] == Total[First/@Split[stc[#]]]+1&]
CROSSREFS
These compositions are counted by A373950.
Positions of ones in A373954.
A003242 counts compressed compositions (or anti-runs).
A114901 counts compositions with no isolated parts.
A116861 counts partitions by compressed sum, by compressed length A116608.
A124767 counts runs in standard compositions, anti-runs A333381.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 encodes compression using compositions in standard order.
A373949 counts compositions by compression-sum.
A373953 gives compression-sum of standard compositions.
Sequence in context: A063963 A101585 A115214 * A143029 A186701 A022123
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 29 2024
STATUS
approved