A319420 Irregular triangle read by rows: row n lists the cuts-resistances of the 2^n binary vectors of length n.

0, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 2, 1, 2, 3, 4, 3, 2, 2, 2, 1, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 5, 4, 3, 3, 3, 2, 2, 3, 3, 2, 1, 2, 2, 2, 3, 4, 4, 3, 2, 2, 2, 1, 2, 3, 3, 2, 2, 2, 3, 3, 3, 4, 5

Offset

0,4

Comments

The cuts-resistance of a vector is defined in A319416. The 2^n vectors of length n are taken in lexicographic order.

Note that here the vectors can begin with either 0 or 1, whereas in A319416 only vectors beginning with 1 are considered (since there we are considering binary representations of numbers).

Conjecture: The row sums, halved, appear to match A189391.

Links

Table of n, a(n) for n=0..63.

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. See table on page 4.

Example

Triangle begins:
0,
1,1,
2,1,1,2,
3,2,1,2,2,1,2,3,
4,3,2,2,2,1,2,3,3,2,1,2,2,2,3,4,
5,4,3,3,3,2,2,3,3,2,1,2,2,2,3,4,4,3,2,2,2,1,2,3,3,2,2,2,3,3,3,4,5,
...

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Table[degdep[Rest[IntegerDigits[n, 2]]], {n, 0, 50}] (* Gus Wiseman, Nov 25 2019 *)

Crossrefs

Keeping the first digit gives A319416.

Positions of 1's are the terms > 1 of A061547 and A086893, all minus 1.

The version for runs-resistance is A329870.

Compositions counted by cuts-resistance are A329861.

Binary words counted by cuts-resistance are A319421 or A329860.

Cf. A000975, A027383, A189391, A318921, A318928, A319411, A329767, A329862, A329865.

Sequence in context: A014600 A165475 A341456 * A267134 A098280 A005793

Adjacent sequences: A319417 A319418 A319419 * A319421 A319422 A319423

Keyword

nonn,tabf,more

Author

N. J. A. Sloane, Sep 22 2018

Status

approved