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

0,3

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.

Links

Table of n, a(n) for n=0..77.

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

Formula

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

Example

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

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}]

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.

Sequence in context: A127800 A035692 A308654 * A334209 A143613 A208955

Adjacent sequences: A329857 A329858 A329859 * A329861 A329862 A329863

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 23 2019

Status

approved