Post making contest 2.0

Known good rule five letter distributions:

13
12,0,1
13,1
12,5,1
12,1,0,1,0,0,1
8,1,4,0,1

These are number of singletons, number of doublets, number of triplets, etc.

A list of better rule 5 distributions, because they can also pass rule 3:
[23=314159]
10,2,3
10,1,2,0,1
11,2,1,0,1
12,4,1
[25=3141592]
12,5,1
12,1,0,1,0,0,1
[31=31415926]
10,3,2,1,1
[61=3141592653589]
5,6,3,0,3,1,2

Come on guys, you can solve this!