A107907 Numbers having consecutive zeros or consecutive ones in binary representation.

3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77

Offset: 1

Author

Reinhard Zumkeller, May 28 2005

Keywords

• nonn
• base

Comments

Also positive integers whose binary expansion has cuts-resistance > 1. For the operation of shortening all runs by 1, cuts-resistance is the number of applications required to reach an empty word. - Gus Wiseman, Nov 27 2019

Examples

			From _Gus Wiseman_, Nov 27 2019: (Start)
The sequence of terms together with their binary expansions begins:
    3:      11
    4:     100
    6:     110
    7:     111
    8:    1000
    9:    1001
   11:    1011
   12:    1100
   13:    1101
   14:    1110
   15:    1111
   16:   10000
   17:   10001
   18:   10010
(End)
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Union of A003754 and A003714.

Complement of A000975.

Cf. A000120, A007088, A070939, A107909, A329862.

Programs

Mathematica

Select[Range[100],MatchQ[IntegerDigits[#,2],{_,x_,x_,_}]&] (* Gus Wiseman, Nov 27 2019 *)
Select[Range[80],SequenceCount[IntegerDigits[#,2],{x_,x_}]>0&] (* or *) Complement[Range[85],Table[FromDigits[PadRight[{},n,{1,0}],2],{n,7}]] (* Harvey P. Dale, Jul 31 2021 *)

Python

def A107907(n): return (m:=n-2+(k:=(3*n+3).bit_length()))+(m>=(1<Chai Wah Wu, Apr 21 2025

Formula

a(A000247(n)) = A000225(n+2).

a(A000295(n+2)) = A000079(n+2).

a(A000325(n+2)) = A000051(n+2) for n>0.

a(n) = m+1 if m >= floor(2^k/3) otherwise a(n) = m where k = floor(log_2(3*(n+1))) and m = n-2+k. - Chai Wah Wu, Apr 21 2025

Extensions

Offset changed to 1 by Chai Wah Wu, Apr 21 2025