Errata and Addenda: ``There Are More Than 2**(n/17) n-Letter Ternary
By S. B. Ekhad and D. Zeilberger
Journal of Integer Sequences, 98.1.9.
Added March 30, 2001:
wrote a more efficient program (using GAP)
that showed that even with the much larger haystack, in which
the components Brinkhuis pairs are not related, one still gets
the same kind of needles, i.e. 2^(1/17) is best possible
(with this method).
Added June 12, 2001:
pointed out that the definition of Brinkhuis
triple-pair, as stated, is insufficient to guarantee
square-freenes-preservation of the homomorphisms, by
presenting a counterexample
Uwe Grimm's message).
However, this is easily fixed. The first condition for being
a Brinkhuis triple-pair is equivalent to demanding that
for every square-free word of length 2: [a,b] (there are six of them)
the four words
[U or V]_a [U or V]_b
are all square-free.
This condition needs to be replaced by the following condition.
For every square-free word of length 3: [a,b,c] (there are twelve of them)
the eight words
[U or V]_a [U or V]_b [U or V]_c ,
It is readily checked (by hand, or use procedure Images1 in JAN),
that the proposed Brinkhuis triple-pair is indeed one, even in
this new, stronger sense, hence the conclusion of the paper is upheld.
I than Uwe Grimm for his careful reading, and for spotting this