I am trying to understand a twist on the Fibonacci bunnies scenario, where the bunnies die x generations after their birth (where x is a positive integer).
An example is shown here.
I understand the table half-way down the link showing the case of x=2 (rabbits die 2 years later) (the table has 5 columns, where column 1 = "Year", column 2 = "Pairs of Newborn Rabbits", column 3 = "Pairs of Newly Mature Rabbits", column 4= "Pairs of Really Mature Rabbits", column 5 = "Total").
According to this table, the total number of rabbits each year follows the sequence 1, 1, 2, 2, 3, 4, 5, 7, 9, ...
The problem is, if asked on an exam to show the case of x=10, I would need 13 columns (too many) in order to derive the last column (and hence solve the sequence of total bunnies per year).
My question is, is there a more efficient table to use so that one could derive the sequence of total bunnies per year for large values of k?
For instance, a table like this one would be helpful (in answer 2). But I think the numbers of total bunnies may be incorrect (because when I try to reapply this table for x=2, I do not get the sequence from the first link of 1, 1, 2, 2, 3, 4, 5, 7, 9, which I know is correct.)