Another rabbits problem

[jasen]

A man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets two new pair which from the second month on becomes productive?

This problem states several important factors:

•     Rabbits take one month to grow up
•     After they have matured (for one month) it takes a pair of rabbits one more month to produce their first 2 pairs of newly born rabbits.
•     We assume that rabbits never die
•     We assume that whenever a new pair of rabbits is produced, it is always a male and a female
•     We assume that these rabbits live in ideal conditions
•     The problem begins with just one pair of newly born rabbits (a male and a female). Given all

this information, how many pairs of rabbits will there be in one year (12 months)?

[Logophobic]

Spoiler

initial state: 0 adult pairs, 1 newborn pair
after 1 month: 1 adult pair, 0 newborn pairs
after 2 months: 1 adult pairs, 2 newborn pairs
after 3 months: 3 adult pairs, 2 newborn pairs
after 4 months: 5 adult pairs, 6 newborn pairs
after 5 months: 11 adult pairs, 10 newborn pairs
after 6 months: 21 adult pairs, 22 newborn pairs
after 7 months: 43 adult pairs, 42 newborn pairs
after 8 months: 85 adult pairs, 86 newborn pairs
after 9 months: 171 adult pairs, 170 newborn pairs
after 10 months: 341 adult pairs, 342 newborn pairs
after 11 months: 683 adult pairs, 682 newborn pairs
after 12 months: 1365 adult pairs, 1366 newborn pairs
total: 2731 pairs of rabbits after one year

 

[jasen]

@logophobic : Right answer, with very clear explanation. 

Can you see any pattern there ?
Can we generalize the problem, such as if a pair begets 3,4, or any number new pairs ....   
or any pairs begets any number new pairs,....

clue : generalization of fibbonacci number.

 

[jasen]

If we make sequences of the numbers, we  get this pattern :

Month	0	1	2	3	4	5	6	7	8	9	10	11	12
sequence	1	1	3	5	11	21	43	85	171	341	683	1365	2731

This pattern is called Jacobsthal numbers, Like Fibonacci numbers,  sequence follow the rule :

The first two numbers in the  sequence are  0 and 1, then each following number is found by adding the number before it to twice the number before that.

In math expression : J_n =  2*J_{(n-2) +}  J_(n-1)

The formula to find the x-th number  is  J_n = ((2ⁿ)-(-1)ⁿ) / 3

So after 1 year there are J₁₃ = (2¹³ + 1) / 3 = 2731 pairs of rabbits