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A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.

%I A329866 #7 Nov 24 2019 10:00:27
%S A329866 1,3,16,30,33,48,55,56,59,60,67,68,72,79,95,97,110,112,118,120,121,
%T A329866 125,134,135,137,143,145,158,160,195,196,219,220,225,231,241,250,258,
%U A329866 270,280,286,291,292,315,316,351,381,382,390,391,393,399,415,416,431,432
%N A329866 Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.
%C A329866 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 A329866 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 A329866 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329866 The sequence of terms together with their binary expansions begins:
%e A329866     1:         1
%e A329866     3:        11
%e A329866    16:     10000
%e A329866    30:     11110
%e A329866    33:    100001
%e A329866    48:    110000
%e A329866    55:    110111
%e A329866    56:    111000
%e A329866    59:    111011
%e A329866    60:    111100
%e A329866    67:   1000011
%e A329866    68:   1000100
%e A329866    72:   1001000
%e A329866    79:   1001111
%e A329866    95:   1011111
%e A329866    97:   1100001
%e A329866   110:   1101110
%e A329866   112:   1110000
%e A329866   118:   1110110
%e A329866   120:   1111000
%e A329866 For example, 79 has runs-resistance 3 because we have (1001111) -> (124) -> (111) -> (3), while the cuts-resistance is 4 because we have (1001111) -> (0111) -> (11) -> (1) -> (), so 79 is in the sequence.
%t A329866 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329866 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329866 Select[Range[100],runsres[IntegerDigits[#,2]]-degdep[IntegerDigits[#,2]]==-1&]
%Y A329866 Positions of -1's in A329867.
%Y A329866 The version for runs-resistance equal to cuts-resistance is A329865.
%Y A329866 Compositions with runs-resistance equal to cuts-resistance are A329864.
%Y A329866 Compositions with runs-resistance = cuts-resistance minus 1 are A329869.
%Y A329866 Runs-resistance of binary expansion is A318928.
%Y A329866 Cuts-resistance of binary expansion is A319416.
%Y A329866 Compositions counted by runs-resistance are A329744.
%Y A329866 Compositions counted by cuts-resistance are A329861.
%Y A329866 Binary words counted by runs-resistance are A319411 and A329767.
%Y A329866 Binary words counted by cuts-resistance are A319421 and A329860.
%Y A329866 Cf. A000975, A003242, A107907, A164707, A329738, A329868.
%K A329866 nonn
%O A329866 1,2
%A A329866 _Gus Wiseman_, Nov 23 2019