Good insight and it may affect gjm’s calculations. It took me enough time to see what you were saying that I’ll spell it out for others...
This string is impossible because it would have stopped at “01”. In general every terminating string of length N will necessarily end with N/2 1′s, and its variable prefix necessarily cannot contain a substring that would have terminated. For example, you can’t count strings that match pattens like: x1xx..., xx11xx..., xxx111xx..., and xxxx1111xx.. and the coverage of later exclusion patterns overlaps with earlier exclusion patterns which makes the counting a bit tricky (that is, 0101110… is doubly excluded by x1 and xxx111).
Surely
is an invalid string?
Good insight and it may affect gjm’s calculations. It took me enough time to see what you were saying that I’ll spell it out for others...
This string is impossible because it would have stopped at “01”. In general every terminating string of length N will necessarily end with N/2 1′s, and its variable prefix necessarily cannot contain a substring that would have terminated. For example, you can’t count strings that match pattens like: x1xx..., xx11xx..., xxx111xx..., and xxxx1111xx.. and the coverage of later exclusion patterns overlaps with earlier exclusion patterns which makes the counting a bit tricky (that is, 0101110… is doubly excluded by x1 and xxx111).
FWIW, my calculations take this into account. (Or at least they’re meant to; I can’t guarantee that I haven’t made any mistakes.)
Sorry, you’re obviously right. I have edited the post to reflect this.