剰余環でフィボナッチ列を作ったらどんな周期が生まれるの？

fib_cycle.rb

# -*- encoding: utf-8 -*-

def fib_cycle(q,a,b)
  fib = [a,b]
  fib_array = [fib]
  while true
    fib = [fib[1], (fib[0] + fib[1]) % q]
    if (index = fib_array.index(fib))
      break
    else
      fib_array << fib
    end
  end
  fib_uniq = fib_array.map{|a| a[0]}.sort.uniq
  p fib_array
  print "fib_cycle(#{q},#{a},#{b})"
  print " => Loop #{fib_array.size}/#{q*q}, Index #{index}"
  #p fib_uniq
  fill_uniq = fib_uniq.size == q ? "!" : ""
  puts " => #{fill_uniq}Unique elems #{fib_uniq.size}/#{q}"
  fib_array
end


#100.times { |i| fib_cycle(10,i/10,i%10) }
#3.upto(100) { |n| fib_cycle(n,1,1) }
total = []

#total += fib_cycle(10,0,0)
#total += fib_cycle(10,0,5)
#total += fib_cycle(10,2,6)
#total += fib_cycle(10,1,8)
#total += fib_cycle(10,2,2)
#total += fib_cycle(10,1,1)
#100.times { |i| p i unless total.include?([i/10,i%10]) }

def fill_fib_cycle(n)
  total = []
  i = 0
  while total.size < n**2
    i += 1 while total.include?([i/n, i%n])
    total += fib_cycle(n, i/n, i%n)
  end
end

fill_fib_cycle(101)