"you'll be able to exclude 13/24 of your gain from taxes"
That's not how it works. The MAXIMUM ($500,000 for married filing jointly) exclusion is reduced . So, if you lived there 13 months, your maximum exclusion is 13/24 x 500,000 = $270,833. So, as long as your gain is less than $270,833, you will be allowed to exclude the entire amount.
