It’s well-known that Fibonacci numbers and the Lucas series are closely related, as shown here.
There’s a lunar connection – the Inex cycle is the cousin of the shorter Saros and lasts nearly 29 years, such that:
18 Inex = 521 years
18 and 521 are both Lucas numbers
What does this tell us?
‘The significance of the inex cycle is not in the prediction, but in the organization of eclipses: any eclipse cycle, and indeed the interval between any two eclipses, can be expressed as a combination of saros and inex intervals. Also when a saros series has terminated, then often one inex after the last eclipse of that saros series, the first eclipse of a new saros series occurs. This in-coming and ex-iting of saros series separated by an interval of 29 years suggested the name for this cycle.’ – Wikipedia [bold added]
Every Lucas number is the sum of two Fibonacci numbers:
18 = 13 + 5
521 = 377 + 144
The number of sidereal months (lunar orbits) in 521 years is 6965.
6965 = 199 x 35
199 is a Lucas number (sum of two Fibonacci numbers: 144 + 55)
521y / 199 = 2.6180904 (~Phi²) = 35 sidereal months
The question then is: why do 18 Inex last 521 years, which is 199 x 35 sidereal months – where 18, 199 and 521 are Lucas numbers and 35 sidereal months is Phi² years?
The chances of this arrangement happening ‘at random’ must be remote, statistically speaking.
Therefore it appears the system is organising itself in this way.
The 521 year period has been called the ‘Hyper Saros’ and even been described as the ‘basic period’ of the solar system by a Belgian Professor.
NB 29 (see earlier quote) is also a Lucas number (sum of Fibonacci 21 + 8).
The Inex cycle is not exactly 29 years because 18 x 29 = 522, not 521.
Actual Inex = 521 / 18 (the two Lucas numbers) = 28.9444~ years.
Nearest Fibonacci equations: 377 / 13 = 29 exactly, 610 / 21 = 29.0476
Wikipedia also notes:
‘Although the inex series lasts much longer than the saros, it is not unbroken: at the beginning and end of a series, eclipses may fail to occur. However once settled down, inex series are very stable and run for many thousands of years.’
Another lunar cycle is the Metonic cycle (~19 years) which has a lesser-known cousin the Callipic cycle (~76 years = 4 x 19). The Greeks used it in their famous Antikythera mechanism. 76 is another Lucas number.
‘This was a more accurate approximation, obtained by taking one day away from every fourth of Meton’s cycles, so creating a 76-year cycle with a mean year of exactly 365.25 days.’
521 / 76 is almost equivalent to the Fibonacci equation 377 / 55 (about 99.98%).
Both are very close to the fourth power of phi i.e. 6.8541.
Numbers fans can pursue all this further here, or at various other sites discussing Lucas and Fibonacci numbers.