A343435 Maximum value of the number of different internal squarefree extensions for words of length n in the set of all finite squarefree words over the alphabet A3 = {1,2,3}.

2, 3, 4, 3, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17

Offset

3,1

Links

Table of n, a(n) for n=3..59.

Jarosław Grytczuk, Hubert Kordulewski, and Bartłomiej Pawlik, Square-free extensions of words, arXiv:2104.04841 [math.CO], 2021. See Table 2 p. 9.

Prog

(Python)
def isf(w): # incrementally squarefree (check factors ending in last letter)
    for l in range(1, len(w)//2 + 1):
        if w[-2*l:-l] == w[-l:]: return False
    return True
def ae(w, sfsnew): # number of internal square free extensions of w
    return len(set(x for x in (w[:i]+c+w[i:] for c in "123" for i in range(1, len(w))) if x in sfsnew))
def aupton(nn):
    alst, sfs = [], set("123")
    for n in range(1, nn+1):
        sfsnew = set(w+c for w in sfs for c in "123" if isf(w+c))
        if n >= 3: alst.append(max(ae(w, sfsnew) for w in sfs))
        sfs = sfsnew
    return alst
print(aupton(30)) # Michael S. Branicky, Aug 31 2021

Crossrefs

Cf. A343436, A343437, A343438.

Sequence in context: A134536 A214551 A371769 * A211010 A131731 A255480

Adjacent sequences: A343432 A343433 A343434 * A343436 A343437 A343438

Keyword

nonn,more

Author

Michel Marcus, Apr 15 2021

Extensions

a(51)-a(59) from Michael S. Branicky, Aug 31 2021

Status

approved