A330085 Length of longest binary word with the property that all distinct occurrences of identical-length blocks agree on at most n positions.

2, 4, 7, 9, 12, 14, 16, 19, 21, 23, 26, 28, 30, 32, 34, 37, 39, 41, 43, 45

Offset

0,1

Comments

By "distinct occurrences" we do not mean that the blocks themselves must be distinct, but rather that they begin at different positions.

Alternatively, this sequence counts the length of the longest binary word w in which each prefix of w matches its corresponding same-length suffix of w in at most n positions.

Links

Table of n, a(n) for n=0..19.

Example

The lexicographically least longest words for n = 0, 1, ..., 10 are as follows:
   0: 01
   1: 0010
   2: 0011010
   3: 001010011
   4: 001101010011
   5: 00011010100110
   6: 0010100110001011
   7: 0011010011101010011
   8: 000110100111010100110
   9: 00100110100011100101011
  10: 01011000111011000101100101
  11: 0001110100100110101011000110
  12: 001010011011000101110001101011

Prog

(Rust) fn max_length(n: u32, l: u32, x: u64) -> u32 {
    (0..2).map(|lowbit| (x << 1) | lowbit)
        .filter(|x| !(n + 1..l + 1).any(|b| (1..l + 1 - b + 1)
                    .any(|occ| (!(x ^ (x >> occ)) & ((1u64 << b) - 1)).count_ones() > n)))
        .map(|x| max_length(n, l + 1, x))
        .max().unwrap_or(l)
}
fn main() {
    for n in 1..64 {
        println!("{} {}", n, (1..=1u64 << (n-1)).map(|x| max_length(n, n, x)).max().unwrap());
    }
} // Falk Hüffner, Jan 31 2020

Crossrefs

Sequence in context: A287074 A304501 A283964 * A175884 A003151 A189939

Adjacent sequences: A330082 A330083 A330084 * A330086 A330087 A330088

Keyword

nonn,more

Author

Jeffrey Shallit, Dec 01 2019

Extensions

a(13)-a(19) from Falk Hüffner, Jan 31 2020

Status

approved