%I #8 Nov 24 2019 10:00:44
%S 0,1,2,7,11,15,18,31,63,75,127,255,511,1023,1234,2047,4095,8191,9638,
%T 16383,32767,65535,131071,262143,524287,1048575,2097151,4194303,
%U 8388607
%N Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.
%C For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined to be the number of applications required to reach a singleton.
%C 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.
%H Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e The sequence of terms together with their binary expansions begins:
%e 0:
%e 1: 1
%e 2: 10
%e 7: 111
%e 11: 1011
%e 15: 1111
%e 18: 10010
%e 31: 11111
%e 63: 111111
%e 75: 1001011
%e 127: 1111111
%e 255: 11111111
%e 511: 111111111
%e 1023: 1111111111
%e 1234: 10011010010
%e 2047: 11111111111
%e 4095: 111111111111
%e 8191: 1111111111111
%e 9638: 10010110100110
%e 16383: 11111111111111
%e 32767: 111111111111111
%e 65535: 1111111111111111
%t runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t das=Table[If[n==0,0,runsres[IntegerDigits[n,2]]-degdep[IntegerDigits[n,2]]],{n,0,1000000}];
%t Table[Position[das,i][[1,1]]-1,{i,First/@Gather[das]}]
%Y Sorted positions of first appearances in A329867.
%Y Compositions with runs-resistance equal to cuts-resistance are A329864.
%Y Runs-resistance of binary expansion is A318928.
%Y Cuts-resistance of binary expansion is A319416.
%Y Compositions counted by runs-resistance are A329744.
%Y Compositions counted by cuts-resistance are A329861.
%Y Binary words counted by runs-resistance are A319411 and A329767.
%Y Binary words counted by cuts-resistance are A319421 and A329860.
%Y Cf. A000975, A003242, A107907, A164707, A329738, A329865, A329866.
%K nonn,more
%O 1,3
%A _Gus Wiseman_, Nov 23 2019