Shell – Cron to not run on specific day but all other days

You can use Anacron for this, it is designed to run jobs at certain intervals without assuming that the system is on continuously. If a job is supposed to be run every month, Anacron will check whether it has run in the previous month, and run it if it hasn't (and remember that it doesn't need to run it again for another month).

In Debian derivatives, Anacron works nicely with Cron to take care of /etc/cron.{daily,weekly,monthly} appropriately (so you simply define your monthly task as a monthly job in /etc/cron.monthly, and it will be run appropriately).

On other systems, you would define your monthly task in /etc/anacrontab.

The crontab(5) man page use a wording that is pretty clear:

Step values can be used in conjunction with ranges. Following a range with "/number" specifies skips of the number's value through the range. For example, "0-23/2" can be used in the hours field to specify command execution every other hour (the alternative in the V7 standard is "0,2,4,6,8,10,12,14,16,18,20,22"). Steps are also permitted after an asterisk, so if you want to say "every two hours", just use "*/2".

The exact wording (and the example) is "skips of the number's value through the range" - and it is implied that it starts at the first number in the range.

This mean if the range is 1-31 for days, the values returned in the case of 1-31/2 or */2 is 1,3,5,7.. etc. This also means that the range is reset to the start value when it has run through.

So you are also correct that in this case, the cronjob would run both on the 31th and 1st the month after.

Please note that cron has 2 fields that are mutually exclusive - the "day of month" and "day of week". So you have to choose one or the other, when running jobs with an interval of days.

If you want to define a cronjob that runs perfectly every other day, you have to use multiple lines and custom define each month according to the current calendar.