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

1,1

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

Links

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

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

Example

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)

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 *)

Prog

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

Crossrefs

Union of A003754 and A003714.

Complement of A000975.

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

Sequence in context: A039093 A379053 A085925 * A080804 A164386 A111909

Adjacent sequences: A107904 A107905 A107906 * A107908 A107909 A107910

Keyword

nonn,base

Author

Reinhard Zumkeller, May 28 2005

Extensions

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

Status

approved