Anti-Fibonacci Sequences and Rings of Saturn
One might assume that anti-Fibonacci number generation is a sufficiently complicated process to be essentially pseudo-random. My back-of-the-envelope calculation went as follows: Let p(i) be the probability that i is anti-Fibonacci. A single existing Fibonacci sequence grows sparser with a rate of phi/i where phi is the base of the golden ratio (sqrt(5) + 1) / 2 with which the Fibonacci values grow, but with a probability of p(i)^2 the i and i+c starts a new Fibonacci sequence and p(i) gets a nudge of 2/i upwards. Putting these in equilibrium and solving for p gives sqrt(sqrt(5) + 1) / 2, or roughly 0.899. I knew I had made some lofty approximations in the derivation above, especially in assuming that the density of multiple Fibonacci sequences would decrease as rapidly as that of one sequence, so I wrote a little program that computes some 100 million first anti-Fibonacci numbers. I was somewhat surprized to see that p is only approximately 0.856 for c = 1. For c = 2 the program gave 0.872, which as again surprizing because I didn't expect c to affect p. But when my computer spat out what looks like interference patterns my jaw really dropped.
Having seen the non-randomness of the average of p over all i and observed how it depends on c, a change in coordinates lead me to wonder if conversely the average of p over c's depends on i. The answer is yes. Below I have a sample of p(i) estimated by c up to ten thousand.
The moral of the story? Whenever playing with number theory, staple your jaw so it won't keep falling off constantly.