expr \( `date +%s` / 604800 \) % 2 >/dev/null || echo "even"
or if you want odd weeks:
expr \( `date +%s` / 604800 + 1 \) % 2 >/dev/null || echo "odd"
There are 604800 seconds in a week.
"% 2" gives the remainder after division by 2.
When expr evaluates to zero the command after the OR (||) is run.
My rubbish collection is on alternate weeks. But cron doesn't know about fortnights and I only put my rubbish out once every two or three months (I don't have much waste). So I use this in cron to trigger an alert on the appropriate day:
0 10 * * mon expr \( `date +\%s` / 604800 + 1 \) \% 2 > /dev/null || alert "garbage day"
I can't remember why I escaped the "%" symbols. I think maybe cron chokes on them if you don't.
Oh, I should add that "alert" is not a standard command. It is a script that I wrote which puts a notice on the screen and uses speech synthesis to announce the message.
(Crossposted from https://miriam-e.dreamwidth.org/332340.html at my Dreamwidth account. Number of comments there so far: )