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Compositions with runs-resistance equal to = cuts-resistance minus 1 are A329869.
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.
allocated for Gus WisemanNumbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.
1, 3, 16, 30, 33, 48, 55, 56, 59, 60, 67, 68, 72, 79, 95, 97, 110, 112, 118, 120, 121, 125, 134, 135, 137, 143, 145, 158, 160, 195, 196, 219, 220, 225, 231, 241, 250, 258, 270, 280, 286, 291, 292, 315, 316, 351, 381, 382, 390, 391, 393, 399, 415, 416, 431, 432
1,2
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.
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.
Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
The sequence of terms together with their binary expansions begins:
1: 1
3: 11
16: 10000
30: 11110
33: 100001
48: 110000
55: 110111
56: 111000
59: 111011
60: 111100
67: 1000011
68: 1000100
72: 1001000
79: 1001111
95: 1011111
97: 1100001
110: 1101110
112: 1110000
118: 1110110
120: 1111000
runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Select[Range[100], runsres[IntegerDigits[#, 2]]-degdep[IntegerDigits[#, 2]]==-1&]
Positions of -1's in A329867.
The version for runs-resistance equal to cuts-resistance is A329865.
Compositions with runs-resistance equal to cuts-resistance are A329864.
Compositions with runs-resistance equal to cuts-resistance minus 1 are A329869.
The runs-resistance of the binary expansion is A318928.
The cuts-resistance of the binary expansion is A319416.
Compositions counted by runs-resistance are A329744.
Compositions counted by cuts-resistance are A329861.
Binary words counted by runs-resistance are A319411 and A329767.
Binary words counted by cuts-resistance are A319421 and A329860.
allocated
nonn
Gus Wiseman, Nov 23 2019
approved
editing
allocated for Gus Wiseman
allocated
approved