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A329865 Numbers whose binary expansion has the same runs-resistance as cuts-resistance.

%I A329865 #7 Nov 24 2019 10:00:18
%S A329865 0,8,12,14,17,24,27,28,35,36,39,47,49,51,54,57,61,70,73,78,80,99,122,
%T A329865 130,156,175,184,189,190,198,204,207,208,215,216,226,228,235,243,244,
%U A329865 245,261,271,283,295,304,313,321,322,336,352,367,375,378,379,380,386
%N A329865 Numbers whose binary expansion has the same runs-resistance as cuts-resistance.
%C A329865 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 A329865 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 A329865 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329865 The sequence of terms together with their binary expansions begins:
%e A329865     0:
%e A329865     8:      1000
%e A329865    12:      1100
%e A329865    14:      1110
%e A329865    17:     10001
%e A329865    24:     11000
%e A329865    27:     11011
%e A329865    28:     11100
%e A329865    35:    100011
%e A329865    36:    100100
%e A329865    39:    100111
%e A329865    47:    101111
%e A329865    49:    110001
%e A329865    51:    110011
%e A329865    54:    110110
%e A329865    57:    111001
%e A329865    61:    111101
%e A329865    70:   1000110
%e A329865    73:   1001001
%e A329865    78:   1001110
%e A329865    80:   1010000
%e A329865 For example, 36 has runs-resistance 3 because we have (100100) -> (1212) -> (1111) -> (4), while the cuts-resistance is also 3 because we have (100100) -> (00) -> (0) -> ().
%e A329865 Similarly, 57 has runs-resistance 3 because we have (111001) -> (321) -> (111) -> (3), while the cuts-resistance is also 3 because we have (111001) -> (110) -> (1) -> ().
%t A329865 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329865 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329865 Select[Range[0,100],#==0||runsres[IntegerDigits[#,2]]==degdep[IntegerDigits[#,2]]&]
%Y A329865 Positions of 0's in A329867.
%Y A329865 The version for runs-resistance equal to cuts-resistance minus 1 is A329866.
%Y A329865 Compositions with runs-resistance equal to cuts-resistance are A329864.
%Y A329865 Runs-resistance of binary expansion is A318928.
%Y A329865 Cuts-resistance of binary expansion is A319416.
%Y A329865 Compositions counted by runs-resistance are A329744.
%Y A329865 Compositions counted by cuts-resistance are A329861.
%Y A329865 Binary words counted by runs-resistance are A319411 and A329767.
%Y A329865 Binary words counted by cuts-resistance are A319421 and A329860.
%Y A329865 Cf. A000975, A003242, A107907, A164707, A319420, A329738, A329868.
%K A329865 nonn
%O A329865 1,2
%A A329865 _Gus Wiseman_, Nov 23 2019