A344346 Numbers k which have an odd number of trailing zeros in their binary reflected Gray code A014550(k).

3, 4, 11, 12, 15, 16, 19, 20, 27, 28, 35, 36, 43, 44, 47, 48, 51, 52, 59, 60, 63, 64, 67, 68, 75, 76, 79, 80, 83, 84, 91, 92, 99, 100, 107, 108, 111, 112, 115, 116, 123, 124, 131, 132, 139, 140, 143, 144, 147, 148, 155, 156, 163, 164, 171, 172, 175, 176, 179, 180

Offset: 1

Author

Amiram Eldar, May 15 2021

Keywords

• nonn
• base
• easy

Comments

Numbers k such that A050605(k-1) is odd.

Numbers k such that A136480(k) is even.

The asymptotic density of this sequence is 1/3.

Examples

			3 is a term since its Gray code, 10, has 1 trailing zero, and 1 is odd.
15 is a term since its Gray code, 1000, has 3 trailing zeros, and 3 is odd.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Gray Code.
• Wikipedia, Gray code.
• Index entries for 2-automatic sequences.

Crossrefs

Cf. A014550, A050605, A081706, A136480.

Similar sequences: A001950 (Zeckendorf), A036554 (binary), A145204 (ternary), A217319 (quaternary), A232745 (factorial), A342050 (primorial).

Programs

Mathematica

Select[Range[180], OddQ @ IntegerExponent[# * (# + 1)/2, 2] &]

Python

def A344346(n):
    def f(x):
        c, s = (n+1>>1)+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return (m<<2)-(n&1) # Chai Wah Wu, Jan 29 2025

Formula

a(n) = A081706(n) + 1. - Hugo Pfoertner, May 16 2021