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%I #6 Nov 24 2019 10:00:00
%S 0,0,1,0,3,6,9,22,47,88,179,354,691,1344,2617,5042,9709,18632,35639,
%T 68010,129556
%N Number of compositions of n with cuts-resistance 2.
%C A composition of n is a finite sequence of positive integers summing to n.
%C 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.
%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e The a(2) = 1 through a(7) = 22 compositions (empty column not shown):
%e (1,1) (2,2) (1,1,3) (3,3) (1,1,5)
%e (1,1,2) (1,2,2) (1,1,4) (1,3,3)
%e (2,1,1) (2,2,1) (4,1,1) (2,2,3)
%e (3,1,1) (1,1,2,2) (3,2,2)
%e (1,1,2,1) (1,1,3,1) (3,3,1)
%e (1,2,1,1) (1,2,2,1) (5,1,1)
%e (1,3,1,1) (1,1,2,3)
%e (2,1,1,2) (1,1,3,2)
%e (2,2,1,1) (1,1,4,1)
%e (1,4,1,1)
%e (2,1,1,3)
%e (2,1,2,2)
%e (2,2,1,2)
%e (2,3,1,1)
%e (3,1,1,2)
%e (3,2,1,1)
%e (1,1,2,1,2)
%e (1,1,2,2,1)
%e (1,2,1,1,2)
%e (1,2,2,1,1)
%e (2,1,1,2,1)
%e (2,1,2,1,1)
%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]==2&]],{n,0,10}]
%Y Column k = 2 of A329861.
%Y Compositions with cuts-resistance 1 are A003242.
%Y Compositions with runs-resistance 2 are A329745.
%Y Numbers whose binary expansion has cuts-resistance 2 are A329862.
%Y Binary words with cuts-resistance 2 are conjectured to be A027383.
%Y Cuts-resistance of binary expansion is A319416.
%Y Binary words counted by cuts-resistance are A319421 and A329860.
%Y Cf. A000975, A003242, A032020, A114901, A240085, A261983, A319420, A329738, A329744, A329864.
%K nonn,more
%O 0,5
%A _Gus Wiseman_, Nov 23 2019
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