Dr. Who, Fibonacci’s Rabbits, and the Wasp Apocalypse (Updated)

Chapter XII from Fibonacci’s Liber abaci describes the following scenario:

A certain 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 a new pair which from the second month on becomes productive?

Solving this riddle, of course, yields the famous Fibonacci sequence:

\text{F}_n = 1,1,2,3,5,8,13,21,34,55,89,144,223,377,610,987,1597,2584,4181,6765...

where the nth term in the sequence is the sum of the previous two terms (n-1 and n-2). That is, \text{F}_n = \text{F}_{n-1} + \text{F}_{n-2} where n > 2.

What is much less well known is that a renegade group of rabbits escaped this enclosure and were later captured after trying to overrun a garden owned by a resident in a nearby town. These rabbits, they discovered, have a unique physiology: they procreate at a much faster rate. We’ll call this the “renegade sequence”:

\text{R}_n = 1,2,4,6,15,18,46,50,115,120,....



\text{Figure 1:} g(x) = 0.7131e^{0.5477x}

In the original storyline for “The Ark in Space” (1975), Dr. Who battles the Wirrn, “a wasp creature [that lays] its eggs inside cryo-preserved humans”; it just so happens the Wirrn’s reproductive pattern follows \text{R}_n (but at a much faster rate—in thousands of births per hour), and after traveling back to the twelfth century to recover the rabbits in order to study their anomalous physiology, Dr. Who tries unsuccessfully to unravel the sequence, a necessary step toward predicting the exact date the Wirrns will take over the earth while also determining whether a proposed vaccine is guaranteed to work fast enough to avert human extinction. Unfortunately, everyone dies (including Dr. Who, whose regenerative powers are neutralized by a wasp-like venom), the invading creatures repopulate the earth, and the series ends. (Cooler heads prevailed, though, and the story was rewritten.)

Figure 1 shows \text{R}_n as a function of time (in n hours) plotted with the regression line (red) defined in the caption (R^2 = 0.9829). Before he received the fatal sting, however, Dr. Who realized the renegade sequence can be derived from an underlying sequence, call it \text{S}_n, defined by \text{R}_n = \sum_{i=1}^n \text{S}_{i}, but he was unable to define an equation that calculates \text{R}_n precisely, one that predicts the total growth of Wirrns for any hour n after the initial infection.

Challenge: Find an equation that calculates any n in \text{R}_n by defining a function f : \text{Z}^+_0\to \text{Z} that generates \text{S}_n.

(I developed this sequence from my work on stock-market trends. I’ll post the answer in an update TBD.)

Spoiler Alert

The answer, based on \text{S}_n, is

\text{R}_n = \sum_{i=1}^n \lceil{\frac{x_i}{2}}\rceil^{3-2\bar{x_i}} + 0^{\bar{x_i}}

where \bar{x} \equiv x\pmod{2}.