A330029 Numbers whose binary expansion has cuts-resistance <= 2.

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 21, 22, 25, 26, 37, 38, 41, 42, 43, 44, 45, 50, 51, 52, 53, 74, 75, 76, 77, 82, 83, 84, 85, 86, 89, 90, 101, 102, 105, 106, 149, 150, 153, 154, 165, 166, 169, 170, 171, 172, 173, 178, 179, 180, 181, 202, 203

Offset

1,3

Comments

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.

Also numbers whose binary expansion is a balanced word (see A027383 for definition).

Also numbers whose binary expansion has all run-lengths 1 or 2 and whose sequence of run-lengths has no odd-length run of 1's sandwiched between two 2's.

Links

Table of n, a(n) for n=1..62.

Example

The sequence of terms together with their binary expansions begins:
    0:
    1:        1
    2:       10
    3:       11
    4:      100
    5:      101
    6:      110
    9:     1001
   10:     1010
   11:     1011
   12:     1100
   13:     1101
   18:    10010
   19:    10011
   20:    10100
   21:    10101
   22:    10110
   25:    11001
   26:    11010
   37:   100101
   38:   100110

Mathematica

degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&, q, Length[#]>0&]]-1;
Select[Range[0, 100], degdep[IntegerDigits[#, 2]]<=2&]

Crossrefs

Union of A000975 and A329862.

Balanced binary words are counted by A027383.

Compositions with cuts-resistance <= 2 are A330028.

Cuts-resistance of binary expansion is A319416.

Cf. A027383, A098504, A107907, A164707, A329860, A329861, A329863, A329865.

Sequence in context: A023776 A063037 A286262 * A368841 A201992 A329297

Adjacent sequences: A330026 A330027 A330028 * A330030 A330031 A330032

Keyword

nonn

Author

Gus Wiseman, Nov 27 2019

Status

approved