OFFSET
0,3
COMMENTS
Every sequence can be uniquely split into a sequence of non-overlapping runs. 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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8, corresponding to (2,1,1) -> (2,2) -> (4), has last term a(11) = 8.
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 trajectory 139 -> 138 -> 136 -> 128 ends with a(139) = 128.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Total[2^Accumulate[Reverse[FixedPoint[Total/@Split[#]&, stc[n]]]]/2], {n, 0, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 01 2022
STATUS
approved