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A330028
Number of compositions of n with cuts-resistance <= 2.
1
1, 1, 2, 3, 7, 13, 23, 45, 86, 159, 303, 568, 1069, 2005, 3769, 7066, 13251, 24821, 46482, 86988, 162758
OFFSET
0,3
COMMENTS
A composition of n is a finite sequence of positive integers summing to n.
For the operation of shortening all runs by 1, cuts-resistance is defined to be the number of applications required to reach an empty word.
EXAMPLE
The a(0) = 1 through a(5) = 13 compositions:
() (1) (2) (3) (4) (5)
(1,1) (1,2) (1,3) (1,4)
(2,1) (2,2) (2,3)
(3,1) (3,2)
(1,1,2) (4,1)
(1,2,1) (1,1,3)
(2,1,1) (1,2,2)
(1,3,1)
(2,1,2)
(2,2,1)
(3,1,1)
(1,1,2,1)
(1,2,1,1)
MATHEMATICA
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], degdep[#]<=2&]], {n, 0, 10}]
CROSSREFS
Sum of first three columns of A329861.
Compositions with cuts-resistance 1 are A003242.
Compositions with cuts-resistance 2 are A329863.
Compositions with runs-resistance 2 are A329745.
Numbers whose binary expansion has cuts-resistance 2 are A329862.
Binary words with cuts-resistance 2 are A027383.
Cuts-resistance of binary expansion is A319416.
Binary words counted by cuts-resistance are A319421 or A329860.
Sequence in context: A298339 A091440 A175211 * A075058 A213968 A213967
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 27 2019
STATUS
approved