%I #7 Nov 24 2019 10:00:27

%S 1,3,16,30,33,48,55,56,59,60,67,68,72,79,95,97,110,112,118,120,121,

%T 125,134,135,137,143,145,158,160,195,196,219,220,225,231,241,250,258,

%U 270,280,286,291,292,315,316,351,381,382,390,391,393,399,415,416,431,432

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

%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 1: 1

%e 3: 11

%e 16: 10000

%e 30: 11110

%e 33: 100001

%e 48: 110000

%e 55: 110111

%e 56: 111000

%e 59: 111011

%e 60: 111100

%e 67: 1000011

%e 68: 1000100

%e 72: 1001000

%e 79: 1001111

%e 95: 1011111

%e 97: 1100001

%e 110: 1101110

%e 112: 1110000

%e 118: 1110110

%e 120: 1111000

%e 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 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;

%t degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;

%t Select[Range[100],runsres[IntegerDigits[#,2]]-degdep[IntegerDigits[#,2]]==-1&]

%Y Positions of -1's in A329867.

%Y The version for runs-resistance equal to cuts-resistance is A329865.

%Y Compositions with runs-resistance equal to cuts-resistance are A329864.

%Y Compositions with runs-resistance = cuts-resistance minus 1 are A329869.

%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, A329868.

%K nonn

%O 1,2

%A _Gus Wiseman_, Nov 23 2019