Demonstrating recursive Fibonacci calculation visits each node in the sequence the number of times corresponding to its position in the calculation. So fib[n] == fib.calls[max_calculated - n].

fibonacci.rb

class Fibonacci
  attr_reader :fibs, :calls

  def initialize
    @fibs = [0, 1]
    @calls = Hash.new { |h, k| h[k] = 0 }
  end

  def [] position
    calls[position] += 1
    return fibs[position] if fibs[position]
    self[position - 1] + self[position - 2]
  end
end

fib = Fibonacci.new
p fib[30].calls

# {
#   30=>1,
#   29=>1,
#   28=>2,
#   27=>3,
#   26=>5,
#   25=>8,
#   24=>13,
#   23=>21,
#   22=>34,
#   21=>55,
#   20=>89,
#   19=>144,
#   18=>233,
#   17=>377,
#   16=>610,
#   15=>987,
#   14=>1597,
#   13=>2584,
#   12=>4181,
#   11=>6765,
#   10=>10946,
#   9 =>17711,
#   8 =>28657,
#   7 =>46368,
#   6 =>75025,
#   5 =>121393,
#   4 =>196418,
#   3 =>317811,
#   2 =>514229,
#   1 =>832040,
#   0 =>514229
# }