A329868 Sorted positions of first appearances in A329867 (difference between the runs-resistance and the cuts-resistance of binary expansion) of each element in the image.

0, 1, 2, 7, 11, 15, 18, 31, 63, 75, 127, 255, 511, 1023, 1234, 2047, 4095, 8191, 9638, 16383, 32767, 65535, 131071, 262143, 524287, 1048575, 2097151, 4194303, 8388607

Offset

1,3

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

Table of n, a(n) for n=1..29.

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

Example

The sequence of terms together with their binary expansions begins:
      0:
      1:                1
      2:               10
      7:              111
     11:             1011
     15:             1111
     18:            10010
     31:            11111
     63:           111111
     75:          1001011
    127:          1111111
    255:         11111111
    511:        111111111
   1023:       1111111111
   1234:      10011010010
   2047:      11111111111
   4095:     111111111111
   8191:    1111111111111
   9638:   10010110100110
  16383:   11111111111111
  32767:  111111111111111
  65535: 1111111111111111

Mathematica

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
das=Table[If[n==0, 0, runsres[IntegerDigits[n, 2]]-degdep[IntegerDigits[n, 2]]], {n, 0, 1000000}];
Table[Position[das, i][[1, 1]]-1, {i, First/@Gather[das]}]

Crossrefs

Sorted positions of first appearances in A329867.

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

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, A329865, A329866.

Sequence in context: A287107 A308550 A018308 * A184485 A001960 A174008

Adjacent sequences: A329865 A329866 A329867 * A329869 A329870 A329871

Keyword

nonn,more

Author

Gus Wiseman, Nov 23 2019

Status

approved