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It doesn’t look as if PlusMaths is going to post my first answer. Just as well, since it was wrong, so here’s my second attempt.
I printed off the biggest Fibonacci rabbit family tree diagram I could find on the net, showing 10 monthly generations resulting in a population of 55 pairs. I subtracted from each monthly population total those rabbit pairs which can't exist that month if each pair stops breeding after producing 3 litters. I entered the new sequence 1 1 2 3 5 7 11 16 24 35 into OEIS and got (amongst others) A023435 which followed on with 52 76 112 164 241 ...
The new recurrence relation, given by OEIS, is a(n)= a(n1) + a(n2)  a(n5). Note that n=5 is the last index at which the Fibonacci and this new sequence continue to share terms. It marks the third and last time the first rabbit pair produces offspring.
The main problem I can see is OEIS also calls this sequence "Dying rabbits". That seems to negate the question, which assumes all rabbits once born continue to live and figure in the subsequent monthly population totals indefinitely. As I'm sure I only crossed off those which couldn't have been born, not those who simply stopped reproducing, this name appears to be wrong.