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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..20.

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.

Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.

Sequence in context: A298339 A091440 A175211 * A075058 A213968 A213967

Adjacent sequences:  A330025 A330026 A330027 * A330029 A330030 A330031

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Nov 27 2019

STATUS

approved

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Last modified September 20 12:13 EDT 2020. Contains 337264 sequences. (Running on oeis4.)