A329860 Triangle read by rows where T(n,k) is the number of binary words of length n with cuts-resistance k.

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

Offset: 0

Author

Gus Wiseman, Nov 23 2019

Keywords

• nonn
• tabl

Comments

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.

Examples

			Triangle begins:
   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  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  92 276 298 178 110  40  22   4   2
   0   2 124 500 564 432 220 132  44  24   4   2
Row n = 4 counts the following words:
  0101  0010  0001  0000
  1010  0011  0111  1111
        0100  1000
        0110  1110
        1001
        1011
        1100
        1101
		

Links

• Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Crossrefs

Column k = 2 appears to be 2 * A027383.

The version for runs-resistance is A319411 or A329767.

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

The version for compositions is A329861.

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

Programs

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
Table[Length[Select[Tuples[{0,1},n],degdep[#]==k&]],{n,0,10},{k,0,n}]

Formula

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