A004780 Binary expansion contains 2 adjacent 1's.

3, 6, 7, 11, 12, 13, 14, 15, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 35, 38, 39, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 70, 71, 75, 76, 77, 78, 79, 83, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base

Comments

Complement of A003714. It appears that n is in the sequence if and only if C(3n,n) is even. - Benoit Cloitre, Mar 09 2003

Since the binary representation of these numbers contains two adjacent 1's, so for these values of n, we will have (n XOR 2n XOR 3n) != 0, and thus a two player Nim game with three heaps of (n, 2n, 3n) stones will be a winning configuration for the first player. - V. Raman, Sep 17 2012

A048728(a(n)) > 0. - Reinhard Zumkeller, May 13 2014

The set of numbers x such that Or(x,3*x) <> 3*x. - Gary Detlefs, Jun 04 2024

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A005809, A048728, A242408.

Complement: A003714.

Subsequences (apart from any initial zero-term): A001196, A004755, A004767, A033428, A277335.

Programs

Haskell

a004780 n = a004780_list !! (n-1)
a004780_list = filter ((> 1) . a048728) [1..]
-- Reinhard Zumkeller, May 13 2014

Maple

q:= n-> verify([1$2], Bits[Split](n), 'sublist'):
select(q, [$0..200])[];  # Alois P. Heinz, Oct 22 2021

PARI

is(n)=bitand(n,n+n)>0 \\ Charles R Greathouse IV, Sep 19 2012

Python

from itertools import count, islice
def A004780_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:n&(n<<1), count(max(startvalue,1)))
A004780_list = list(islice(A004780_gen(),30)) # Chai Wah Wu, Jul 13 2022

Formula

a(n) ~ n. - Charles R Greathouse IV, Sep 19 2012

Extensions

Offset corrected by Reinhard Zumkeller, Jul 28 2010