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Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.
16

%I #8 Nov 24 2019 00:00:33

%S 1,0,2,0,2,2,0,2,4,2,0,2,8,4,2,0,2,12,12,4,2,0,2,20,22,14,4,2,0,2,28,

%T 48,28,16,4,2,0,2,44,84,70,32,18,4,2,0,2,60,162,136,90,36,20,4,2,0,2,

%U 92,276,298,178,110,40,22,4,2,0,2,124,500,564,432,220,132,44,24,4,2

%N Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.

%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.

%F For positive indices, T(n,k) = 2 * A319421(n,k).

%e Triangle begins:

%e 1

%e 0 2

%e 0 2 2

%e 0 2 4 2

%e 0 2 8 4 2

%e 0 2 12 12 4 2

%e 0 2 20 22 14 4 2

%e 0 2 28 48 28 16 4 2

%e 0 2 44 84 70 32 18 4 2

%e 0 2 60 162 136 90 36 20 4 2

%e 0 2 92 276 298 178 110 40 22 4 2

%e 0 2 124 500 564 432 220 132 44 24 4 2

%e Row n = 4 counts the following words:

%e 0101 0010 0001 0000

%e 1010 0011 0111 1111

%e 0100 1000

%e 0110 1110

%e 1001

%e 1011

%e 1100

%e 1101

%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;

%t Table[Length[Select[Tuples[{0,1},n],degdep[#]==k&]],{n,0,10},{k,0,n}]

%Y Column k = 2 appears to be 2 * A027383.

%Y The version for runs-resistance is A319411 or A329767.

%Y The cuts-resistance of the binary expansion of n is A319416(n).

%Y The version for compositions is A329861.

%Y Cf. A000975, A164707, A261983, A318928, A319420, A319421, A329738, A329865.

%K nonn,tabl

%O 0,3

%A _Gus Wiseman_, Nov 23 2019