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 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.
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 given in row 11 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).
EXAMPLE
Triangle begins:
0
1
2
3 2
4
5
6
7 4
8
9
10 8
11 10 8
12
13
14 10 8
For example, the trajectory of 29 is 29 -> 21 -> 17, corresponding to the compositions (1,1,2,1) -> (2,2,1) -> (4,1).
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
Table[NestWhileList[stcinv[Total/@Split[stc[#]]]&, n, MatchQ[stc[#], {___, x_, x_, ___}]&], {n, 0, 50}]
CROSSREFS
Final terms are A353855.
Counting rows by weight of final term gives A353856.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order.
A318928 gives runs-resistance of binary expansion.
A329739 counts compositions with all distinct run-lengths.
A333627 ranks the run-lengths of standard compositions.
A353932 lists run-sums of standard compositions.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 01 2022
STATUS
approved