Project Euler 197: Investigating the behavior of a recursively defined sequence

Problem Description

Given is the function f(x) = ⌊2^{30.403243784-x2}⌋ × 10^-9 ( ⌊ ⌋ is the floor-function),
the sequence u_n is defined by u₀ = -1 and u_n+1 = f(u_n).

Find u_n + u_n+1 for n = 10¹².
Give your answer with 9 digits after the decimal point.

Analysis

We first made a plot of this function to see what we were dealing with. It seemed to oscillate and converge very quickly bouncing back and forth between 0.6811… and 1.0294… ad infinitum. This means that instead of 10¹² iterations only a few hundred are required.

Project Euler 197 Solution

Afterthoughts

• The step size of 51 was arbitrary and could be any odd number. The bigger the step size the faster the solution (fewer iterations) but the deeper the level of recursion.
  
• Graph of relation via Mathematica showing “ringing bell” pattern