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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.
5

%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