A356677 The lexicographically earliest infinite squarefree sequence of nonnegative integers that starts with 1.

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 2, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 2, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 3, 0, 1, 0, 3, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 0, 2, 3, 0, 1, 0

Offset

1,4

Comments

Conjecture: For any integer k>2, prepending a k at the beginning of this sequence gives the corresponding least word beginning k,1.

After the first 5177 letters, this becomes the sequence A356676.

Links

Joey Lakerdas-Gayle, Table of n, a(n) for n = 1..10000

Siddharth Berera, Andrés Gómez-Colunga, Joey Lakerdas-Gayle, John López, Mauditra Matin, Daniel Roebuck, Eric Rowland, Noam Scully, and Juliet Whidden, The lexicographically least square-free word with a given prefix, arXiv:2210.00508 [math.CO], 2022.

Joey Lakerdas-Gayle, (Python) Modified Main-Lorentz algorithm to more efficiently compute terms

Prog

(Python)
# check if appending letter to the end of word introduces a square
def makes_square(word, letter):
    new_word = word+[letter]
    for l in range(1, len(new_word) // 2 + 1):
        if new_word[-l:] == new_word[-2*l:-l]:
            return True
    return False
# returns a list of the first n letters of L(word)
def build_square_free(word, n):
    new_word = word[:]
    for i in range(n-len(word)):
        next_letter = 0
        while makes_square(new_word, next_letter):
            next_letter += 1
        new_word += [next_letter]
    return new_word
# returns a list of the first n terms of A356677
def A356677_list(n):
    return build_square_free([1], n)
(Python) # See link to more efficient algorithm

Crossrefs

Cf. A356676 (suffix).

Other word starts: A007814 (w=0), this sequence (w=1), A356678 (w=2), A356679 (w=3), A356680 (w=1,2), A356681 (w=1,3), A356682 (w=2,1).

Sequence in context: A129688 A356678 A356680 * A309011 A365921 A086713

Adjacent sequences: A356674 A356675 A356676 * A356678 A356679 A356680

Keyword

nonn

Author

Joey Lakerdas-Gayle, Aug 22 2022

Extensions

Name edited by N. J. A. Sloane, Nov 26 2022

Status

approved