A356679 The lexicographically earliest infinite squarefree sequence of nonnegative integers that starts with 3.

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

Offset

1,1

Comments

For any integer k>2, prepending a k at the beginning of this sequence gives the corresponding least word beginning k,3.

After the first 14070 letters, this becomes the image of the sequence A356676 under the ruler morphism, n->0(n+1).

From Jianing Song, Nov 28 2022: (Start)

The lexicographically earliest infinite squarefree word starting with 2,3 is not obtained by prepending 2 to the sequence: denote the corresponding sequence by {b(n)}. We have b(n) = a(n-1) for 2 <= n <= 521, but then we have a(1..260) = a(262..521) and a(261) = 2, which means that b(522) > a(521) (otherwise b(1..261) = b(262..522)). Actually, a(521) = 0 and b(522) = 1.

a(3768) is the first term equal to 4, and a(4768) is the first term equal to 5. (End)

Links

Jianing Song, 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.

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 A356679
def A356679_list(n):
    return build_square_free([3], n)
(PARI) issquareword(v) = my(n=#v); for(i=1, n\2, if(v[n-2*i+1..n-i] == v[n-i+1..n], return(1))); return(0)
the_first_N_terms(N) = my(v=vector(N)); v[1]=3; for(n=2, N, for(k=0, oo, v[n]=k; if(!issquareword(v[1..n]), break()))); v \\ Jianing Song, Nov 28 2022

Crossrefs

Cf. A356676.

Other word starts: A007814 (w=0), A356677 (w=1), A356678 (w=2), this sequence (w=3), A356680 (w=1,2), A356681 (w=1,3), A356682 (w=2,1).

Sequence in context: A331569 A341716 A216600 * A356681 A093684 A101270

Adjacent sequences: A356676 A356677 A356678 * A356680 A356681 A356682

Keyword

nonn

Author

Joey Lakerdas-Gayle, Oct 18 2022

Extensions

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

Status

approved