OFFSET
0,3
COMMENTS
We define a partial run of a sequence to be any contiguous constant 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.
The term rucksack is short for run-knapsack.
EXAMPLE
The terms together with their corresponding compositions begin:
0: ()
1: (1)
2: (2)
3: (1,1)
4: (3)
5: (2,1)
6: (1,2)
7: (1,1,1)
8: (4)
9: (3,1)
10: (2,2)
12: (1,3)
13: (1,2,1)
15: (1,1,1,1)
Missing are:
11: (2,1,1)
14: (1,1,2)
23: (2,1,1,1)
27: (1,2,1,1)
29: (1,1,2,1)
30: (1,1,1,2)
39: (3,1,1,1)
43: (2,2,1,1)
46: (2,1,1,2)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Select[Range[0, 100], UnsameQ@@Total/@Union@@Subsets/@Split[stc[#]]&]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 15 2022
STATUS
approved