# Help with formula for complicated pattern

This picture represents 56 Lunar years.

On the Left side, you have the months, not twelve but 13 ( you can go here for an explanation on that) - The numbers highlighted in orange above all the numbers in blue, green and turquoise, are years.

This is an excel sheet I made - It represents 56 Lunar years and 693 months. All of those numbers in the blue, green and turquoise, represent Saturdays.

Within this you have an eclipse that will happen every 220 months, the first one on the first day of the fourth month of the first year; 220 months later, it will be on the first day of the second month of the nineteenth year.

My question regards finding an equation that will help me find the eclipses for a given 56 year cycle - let me explain more..

This image is one 56 year cycle - each 56 year cycle has 3 eclipses max, because they happen every 18 solar years and 10 days OR 220 Lunar Months.

The pattern will always be offset until it resets at some point and starts over. What I mean by this, is that: each new 56 year cycle, the first ecplipse for that cycle will be on a different day, until eventually, after $x$ amount of cycles (i think that number is $63$) it will 'reset', or start back over at the beginning, where the first eclipse for that 56 year cycle will be on the fourth month of the first year. (with this, i think it will be safe to say that we will have $63$ (for example) 56 year cycles... )

How, or where, can I begin to find a solution to this. I figured asking here would be my best option, because there are A LOT of talented mathematicians here. Any help is appreciated.

If you would like further explanation, please ask - this all seems clear in my head, but I am not so sure it will be that clear in yours.

thank you...

p.s. I am currently working on finding when the fourth eclipse will happen... this is what I have so far...

• Define "eclipse." Lunar eclipse or solar eclipse? Total (annular) or partial? – scott Mar 24 '17 at 16:04
• It is a solar eclipse – Carlos Carlsen Mar 24 '17 at 16:57
• Niether "18 solar years plus 10 days", nor "220 lunar months" defines a set period of time. The first specifies a period of either 6574 or 6575 days, depending on how the period aligns with the calendar. 220 Lunar months will be 107 29-day months and 113 30-day months or 106 29-day months and 114 30-day months, totalling 6493 or 6494 days respectively. Thus you are offering 4 different possible intervals with no information on how to choose which interval occurs at any given time. – Paul Sinclair Mar 25 '17 at 13:33