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Python

# see link for faster, bit-based version
from itertools import product
def is_strm(w): # w is a finite Sturmian word
    for l in range(2, len(w)):
        facts = set(w[j:j+l] for j in range(len(w)-l+1))
        counts = set(f.count('0') for f in facts)
        if max(counts) - min(counts) > 1:
            return False
    return True
def is_clsd(w): # w is a closed word
    if len(set(w)) <= 1: return True
    for l in range(1, len(w)):
        if w[:l] == w[-l:] and w[:l] not in w[1:-1]:
            return True
    return False
def is_cs(w):   # w is a closed Sturmian word
    return is_clsd(w) and is_strm(w)
def a(n):
    return 2*sum(is_cs("0"+"".join(b)) for b in product("01", repeat=n-1))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Sep 27 2021