A319420 - OEIS

%I #15 Nov 27 2019 20:35:30

%S 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,

%T 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

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

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

%C 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).

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

%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, 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 Apparently unpublished.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.

%e Triangle begins:

%e 0,

%e 1,1,

%e 2,1,1,2,

%e 3,2,1,2,2,1,2,3,

%e 4,3,2,2,2,1,2,3,3,2,1,2,2,2,3,4,

%e 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,

%e ...

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

%t Table[degdep[Rest[IntegerDigits[n,2]]],{n,0,50}] (* _Gus Wiseman_, Nov 25 2019 *)

%Y Keeping the first digit gives A319416.

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

%Y The version for runs-resistance is A329870.

%Y Compositions counted by cuts-resistance are A329861.

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

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

%K nonn,tabf,more

%O 0,4

%A _N. J. A. Sloane_, Sep 22 2018