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A329866
Numbers whose binary expansion has its runs-resistance equal to its cuts-resistance minus 1.
5
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
OFFSET
1,2
COMMENTS
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.
LINKS
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.
EXAMPLE
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
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.
MATHEMATICA
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&]
CROSSREFS
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 = cuts-resistance minus 1 are A329869.
Runs-resistance of binary expansion is A318928.
Cuts-resistance of 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.
Sequence in context: A258717 A013196 A371382 * A339634 A377084 A196264
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 23 2019
STATUS
approved