Helios has suggested a calendar in which each year has 198 parts and each lunar month has 16 parts and once every nodetide the calendar is corrected by removing one part from the year to form a short 197-part year. Such a calendar would give the number E of eclipse seasons in Y years of M months as
E = 200*Y - 16*M.
I show this next.
Noting that the number N of nodetides in Y years of E eclipse seasons is given by
N = E - 2*Y
E = N + 2*Y
Substituting this into my formula (shown 1st) gives
N = 198*Y - 16*M
In a cycle of Y years and M months, we'd have 16*M = 198*Y - K parts, where K is number of the corrections. We then get
K = 198*Y - 16*M = N
Therefore correcting once every nodetide, does implement my formula for eclipse seasons.
I don't know how Helios derived his year of 365.242715 days. Perhaps, it is the mean year for which my formula is most accurate. A year whose mean year is considerably less would require an additional correction once every few thousand years. For example, a mean year of 365.242215 days would require an additional correction about once every 2000 years. The 4160-year cycle would require 448 corrections one of which must be an additional correction to give it 447 nodetides.
K = 198*Y - 16*M = N + A, where A is the number of additional corrections.
These cycles with number of nodetides in () require no additional corrections:
1135(122), 1154(124), 1173(126),
1526(164), 1545(166), 1564(168),