I was reading about a calendar system with these characteristics:
Years are grouped into cycles which begin with four normal years after which every fourth subsequent year in the cycle is a leap year. Cycles are grouped into grand cycles of either 128 years (composed of cycles of 29, 33, 33, and 33 years) or 132 years, containing cycles of of 29, 33, 33, and 37 years. A great grand cycle is composed of 21 consecutive 128 year grand cycles and a final 132 grand cycle, for a total of 2820 years. reference
If you write down years vertically separated in cycles you will have something like this:
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++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
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++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
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++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
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++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
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++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
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**** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** ****
++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
**** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** ****
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**** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** **** ****
++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++ ++++
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+++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++ +++
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Here, grand cycles are separated by one space and leap years showed by + and normal years by *.
The problem is finding number of passed leap years in current period (2820 years) for given year. Not a hard problem but I surprised when saw author used a very simplified formula for this calculation (With familiar numbers 682 and 2816):
$$\lfloor\frac{year\cdot682 - 110}{2816}\rfloor$$
I searched and found dozens of code using same trick which is probably came from this Lisp implementation (lines #19 to #21 in fixed-from-arithmaric-persian function). But I couldn't understand why this holds true. Why?