A329866 Numbers 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

Offset: 1

Author

Gus Wiseman, Nov 23 2019

Keywords

• nonn

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.

Examples

			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.
		

Links

• Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

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.

Cf. A000975, A003242, A107907, A164707, A329738, A329868.

Programs

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&]