So by banks logic "in a year" means in 360 days...and it means that 6 more days are coming to finish a calendar year.

After a year: 100 -> 100,98 (100 * 1,0098 or 100 + 100 * 0,98 / 100 )

so until that you compound annually

361 / 100,9827489 362 / 100,9854978...

and then daily...

Try to compound always the same way.

Hello, I'm confused... and can't find my fault - please help!

As an example you give "date_nb_days(100, 101, 0.98) --> "2017-01-01" (366 days)".
After a year: 100 -> 100,98 (100 * 1,0098 or 100 + 100 * 0,98 / 100 ) and 360 days.
InterestsPerDay now: 100,98 * 0,98 / 36000 = 0,0027489.
days / sum
361 / 100,9827489
362 / 100,9854978
363 / 100,9882467
364 / 100,9909956
365 / 100,9937445
366 / 100,9964934
367 / 100,9992423
368 / 101,0019912

other try:
0,02 / 0,0027489 = 7,2756375277383680745025282840409

Can you please explain, how you came to 366 days?

I think it may be helpful if you re-word the description, because it implies that the amount a0 is deposited every year. I would suggest that you phrase it as "You have an amount of money a0 > 0 and you deposit it on the 1st of January 2016, with an annual interest rate of p% > 0" This should help clear up the ambiguity in the first sentence. The reader can discern the intended meaning currently, but only because the date includes the year and thus an annual deposit wouldn't make sense. Also, you should add "compounded annually" or "compounded daily" to the end of that first sentence. Banks traditionally tell customers the interest rate in the form of the APY, which is annual, but the real interest rate is continuously compounded. So, it may be easier if you explicitly state how often it is compounded, because our banking conventions have separated the compounding frequency from the advertised "annual" rate.

If daily rate is p/36000 than how can we get a right amount in 366 days with a daily added rate? (read "360 is not equal to 366").

How is the interest added -daily, monthly, quarterly or yearly?