A329870 Runs-resistance of the binary expansion of n without the first digit.

0, 0, 1, 2, 2, 1, 1, 3, 2, 3, 3, 2, 3, 1, 1, 3, 4, 2, 4, 2, 3, 3, 3, 3, 2, 4, 2, 4, 3, 1, 1, 3, 4, 3, 3, 4, 4, 3, 4, 5, 2, 4, 4, 5, 3, 3, 3, 3, 5, 4, 4, 2, 5, 4, 3, 4, 4, 3, 3, 4, 3, 1, 1, 3, 4, 3, 3, 4, 3, 2, 3, 3, 4, 4, 2, 3, 3, 3, 4, 5, 4, 3, 4, 2, 5, 4

Offset

2,4

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.

Links

Table of n, a(n) for n=2..87.

Example

Minimal representatives with each image are:
    2: (0)
    4: (0,0) -> (2)
    5: (0,1) -> (1,1) -> (2)
    9: (0,0,1) -> (2,1) -> (1,1) -> (2)
   18: (0,0,1,0) -> (2,1,1) -> (1,2) -> (1,1) -> (2)
   41: (0,1,0,0,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2)
  150: (0,0,1,0,1,1,0) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1) -> (2)

Mathematica

Table[Length[NestWhileList[Length/@Split[#]&, Rest[IntegerDigits[n, 2]], Length[#]>1&]]-1, {n, 2, 100}]

Crossrefs

Keeping the first digit gives A318928.

Cuts-resistance is A319420.

Compositions counted by runs-resistance are A329744.

Binary words counted by runs-resistance are A319411 and A329767.

Cf. A107907, A319416, A329860, A329861, A329865, A329867.

Sequence in context: A047110 A288533 A093869 * A057431 A179541 A057060

Adjacent sequences: A329867 A329868 A329869 * A329871 A329872 A329873

Keyword

nonn

Author

Gus Wiseman, Nov 25 2019

Status

approved