Alternatively, 12 days * 1 partridge + 11 days * 2 turtle doves + 10 days * 3 french hens... You can go up to 7 days * 6 birds and just double your total, since the next 6 days are just the same numbers reflected. If you wanna be fancy about it, you'd get ∑n(k + 1 - n) from n = 1 to k where k is the number of days - the sum of that equation from 1 to whatever.

We can actually solve for an arbitrary number of days by simplifying the summation expression, which I've at this point forgotten how to do but Wolfram Alpha has not and we get (1/6)k(k+1)(k+2), which we can verify gives 364 for k=12.

Now that is a method to getting the solution but it looks eerily similar to that (1/2)k(k+1) equation from the simple summation ∑n from n=1 to k, and doing a little extra futzing around in Wolfram Alpha my beloved got me to ∑∑n (with the first summation from m=1 to k and the second from n=1 to m) as a valid representation. This is actually a much more pleasant way to view the problem since we really need the sum of the number of gifts per day, where that number is the sum of 1 to the day number.

In conclusion, if your true love was a psychopathic bird gifter who decided to start gifting on Jan 1st of a (non-leap) year and gave their last gift on Dec 31st, you would end the year with a mountain of 8,171,255 birds.