- #1
Jim01
- 36
- 0
Homework Statement
A single pair of rabbits (male and female) is born at the beginning of a year. Assume the following conditions:
(1) Rabbit pairs are not fertile during their first two months of life, but thereafter give birth to three new male/female pairs at the end of every month.
(2) No rabbits die
(a) Let Sn = the number of pairs of rabbits alive at the end of month n, for each ionterger n>=1, and let S0 = 1. Find a recurrence relation for S0, S1, S2, ...
Homework Equations
Original Fibonacci equation = Fn = Fk-1 + Fk - 2, where F0 = 1 and F1 = 1.
The Attempt at a Solution
I have drawn a genealogy chart to the 7th generation and have come up with
S0 = 1,
S1 = 1,
S2 = 1,
S3 = 4,
S4 = 7,
S5 = 10,
S6 = 22,
S7 = 43,
The problem is that I cannot figure out a way to come up with the equation which would give me the recurrence relation.
I tried doing Fn = Fk-1 + Fk - 2 + Fk - 3 +1, but that doesn't work unless n >= 3 and does not work past S4. I've tried several other combinations as well but I can't figure it out.
Is this one of those problems where you just have to "see" the answer or is there a procedure I can use to get it?