365.2421896698  0.00000615359 T  7.29E10 T^{2} + 2.64E10 T^{3} (days)
where T = (JD  2451545.0) / 36525 and JD is the Julian day number. The value of the mean tropical year for the epoch of J2010.0 (Julian day of 2455197.5, and thus T = 0.1) was calculated with Laskar's equation above.All the progressive solar calendars since Old Egyptian times are arithmetical calendars. This means an easy rule to try to reach the best possible astronomical value.
In the history of solar calendars notably these five
rules (approximations) shown below were used, are used, or are proposed.
Calendar rule 


Old Egyptian  365  = 365. 000 000 000 
Julian  365 + ¼  = 365. 250 000 000 
Gregorian  365 + 97/400  = 365. 242 500 000 
Khayyam  365 + 8/33  = 365. 24 24 24 24 
Revised Julian  365 + 218/900  = 365. 24 22 22 22 
von Mädler  365 + 31/128  = 365. 242 187 500 
Mean tropical year at epoch J2010.0  = 365. 242 189 054 433 974 
The currently widelyused Gregorian calendar has an average year of:
However, according to von Mädler, if the years
3200, 6400, 9600, 12,800,
16,000, and so on are NOT
leap years (an additional rule to the ones already in use with the
Gregorian calendar),
the duration of the mean Gregorian
year will then be 365.2421875 days,
and
this
approaches
very
close
the
real
duration
of the mean tropical
year which lasts 365.24219 days.
This
additional
rule
will
produce
a
very
little
error of only
3.125  3.100 = 0.025 days in 10,000 years.
I had figured out a better system than the Gregorianvon Mädler one for leap year
decisions, but I plan on using it only if the Cesidian
calendar becomes adopted worldwide.
HMRD Cesidio Tallini