A295897 Numbers in whose binary expansion there are no 1-runs of odd length followed by a 0 to their right.

0, 1, 3, 6, 7, 12, 13, 15, 24, 25, 27, 30, 31, 48, 49, 51, 54, 55, 60, 61, 63, 96, 97, 99, 102, 103, 108, 109, 111, 120, 121, 123, 126, 127, 192, 193, 195, 198, 199, 204, 205, 207, 216, 217, 219, 222, 223, 240, 241, 243, 246, 247, 252, 253, 255, 384, 385, 387, 390, 391, 396, 397, 399, 408, 409, 411, 414, 415, 432

Offset: 1

Author

Antti Karttunen, Dec 01 2017

Keywords

• nonn
• base

Comments

No runs of 1-bits of odd length allowed in the binary expansion of n (A007088), except that when n is an odd number, then the rightmost run may have an odd length. Subsequence A277335 does not allow that exception.

A005940(1+a(n)) yields a permutation of A028982, squares and twice squares.

Running maximum without repetition of the decimal equivalent of Gray code for n (A003188). - Frédéric Nouvier, Aug 14 2020

Links

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

Crossrefs

Cf. A005940, A028982, A280873.

Subsequence of A004760.

Cf. A277335 (a subsequence).

Cf. A295896 (characteristic function).

Programs

Python

[x ^ (x>>1) for x in range(0,2048) if (x & (x<<1) == 0)]
# Frédéric Nouvier, Aug 14 2020

Python

def A295897(n):
    tlist, s = [1,2], 0
    while tlist[-1]+tlist[-2] <= n: tlist.append(tlist[-1]+tlist[-2])
    for d in tlist[::-1]:
        s <<= 1
        if d <= n:
            s += 1
            n -= d
    return s>>1^s # Chai Wah Wu, Apr 25 2025

Rust

fn main() {
    for i in (0..2048)
        // Filter to get A003714
        .filter(|n| n & (n << 1) == 0)
        // Map to produce A295897
        .map(|n| n ^ (n >> 1))
    {
        println!("{}", i);
    }
} // Frédéric Nouvier, Aug 14 2020

Formula

a(n) = A003714(n-1) XOR ( A003714(n-1) >> 1 ). - Frédéric Nouvier, Aug 14 2020