A173022 Number of numbers <= n whose binary representation is without isolated ones.

1, 1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 9, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21

Offset

0,4

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Formula

a(A144795(n+1)) = a(A144795(n)) + 1.

a(2^n - 1) = A005251(n+2).

A173021(n) <= a(n) <= A173023(n).

Example

a(20) = #{0,3,6,7,12,14,15} = #{0,11,110,111,1100,1110,1111} = 7.

Mathematica

Accumulate[Array[Boole[FreeQ[Split[IntegerDigits[#, 2]], {1}]] &, 100, 0]] (* Paolo Xausa, Oct 15 2024 *)

Prog

(Python)
from sympy import Matrix
def A173022(n):
    s = bin(n)[-1:1:-1]
    t = '0'+s+'0'
    return sum((Matrix([[2, -1, 1], [1, 0, 0], [0, 1, 0]])**i*Matrix([1, 1, 0]))[0] for i in range(len(s)) if s[i]=='1' and not '010' in '0'+t[i+2:])+('010' not in t) # Chai Wah Wu, Jun 10 2026

Crossrefs

Cf. A005251, A007088, A144795, A173021, A173023.

Sequence in context: A228482 A091822 A358178 * A060973 A352228 A097915

Adjacent sequences: A173019 A173020 A173021 * A173023 A173024 A173025

Keyword

base,nonn

Author

Reinhard Zumkeller, Feb 07 2010

Status

approved