cp's OEIS Frontend

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

%I A319420 #15 Nov 27 2019 20:35:30
%S A319420 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 A319420 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 A319420 Irregular triangle read by rows: row n lists the cuts-resistances of the 2^n binary vectors of length n.
%C A319420 The cuts-resistance of a vector is defined in A319416. The 2^n vectors of length n are taken in lexicographic order.
%C A319420 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 A319420 Conjecture: The row sums, halved, appear to match A189391.
%H A319420 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 Apparently unpublished.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 A319420 Triangle begins:
%e A319420 0,
%e A319420 1,1,
%e A319420 2,1,1,2,
%e A319420 3,2,1,2,2,1,2,3,
%e A319420 4,3,2,2,2,1,2,3,3,2,1,2,2,2,3,4,
%e A319420 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 A319420 ...
%t A319420 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A319420 Table[degdep[Rest[IntegerDigits[n,2]]],{n,0,50}] (* _Gus Wiseman_, Nov 25 2019 *)
%Y A319420 Keeping the first digit gives A319416.
%Y A319420 Positions of 1's are the terms > 1 of A061547 and A086893, all minus 1.
%Y A319420 The version for runs-resistance is A329870.
%Y A319420 Compositions counted by cuts-resistance are A329861.
%Y A319420 Binary words counted by cuts-resistance are A319421 or A329860.
%Y A319420 Cf. A000975, A027383, A189391, A318921, A318928, A319411, A329767, A329862, A329865.
%K A319420 nonn,tabf,more
%O A319420 0,4
%A A319420 _N. J. A. Sloane_, Sep 22 2018