The Fibonacci sequence is the sequence of whole numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. Combinations of the numbers in this sequence occur in nature, see page 338.

• Let us present these Fibonacci numbers as a function F(n). When we put the function in a table, the result looks like

  n F(n)

  0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 .... 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 ....

  So F(3) = 2 and F(11) = 89.
• So far we understand the definition of the sequence of Fibonacci numbers somewhat informally. An example of a precise mathematical definition is:
  • F(0) = 0 and F(1) = 1,    the initial conditions part of the definition.
  • F(n) = F(n-1) + F(n-2), for all n greater than or equal 2    the recursive part of the definition.
  With these conditions, the Fibonacci sequence is completely and uniquely determined.

• Following the book, consider the growth rate of the terms of the Fibonacci sequence. It lists
  • F(2) / F(1) = 1/1 = 1
  • F(3) / F(2) = 2/1 = 2
  • F(4) / F(3) = 3/2 = 1.5
  • F(5) / F(4) = 5/3 = 1.66666667 approximately
  • F(6) / F(5) = 8/5 = 1.6
  • F(7) / F(6) = 13/8 = 1.625
  • F(8) / F(7) = 21/13 = 1.61538462 approximately
  As this pattern suggests, the consecutive quotients approach some constant number we call φ (Greek letter phi) or G, for Golden Ratio. When we keep extending the table above by more quotients, we see that
  • φ = G = 1.618033988749894848204586834 approximately
  When we stop our investigations just because we can approximate φ with arbitrary precision, we potentially lose new patterns. So we search for equations involving φ = G. One easily shows that the quotients in the list above can also be written as
  • 1/1 = 1
  • 2/1 = 1 +1/1 = 1 + 1
  • 3/2 = 1 +1/2 = 1 + 1/(1 + 1)
  • 5/3 = 1 +2/3 = 1 + 1/(1 + 1/(1 + 1))
  • 8/5 = 1 +3/5 = 1 + 1/(1 + 1/(1 + 1/(1 + 1)))
  • 13/8 = 1 +5/8 = 1 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + 1))))
  These nested quotients have a convergence property by which it is reasonable to write
  • G = 1 + 1/G   (or: φ = 1 + 1/φ)
  Multiply both sides by G. So G^2 = G + 1. So G^2 - G - 1 = 0. When we solve this quadratic equation, we get roots (1 + 5^(1/2)) / 2 and (1 - 5^(1/2)) / 2. The correct solution is
  • φ = G = (1 + 5^(1/2)) / 2
  Note the presence of 5^(1/2), also written sqrt(5), the square root of 5. The number ψ (Greek letter psi) or H = (1 - 5^(1/2))/2 is such that G+H = 1, so ψ = H = 1-G = 1 - φ, so
  • ψ = H = -0.618033988749894848204586834 approximately
• The Golden Ratio φ = G and the associated number ψ = H give two sequences that behave like the Fibonacci sequence:

  n G^n

  0 1 2 3 4 5 6 7 8 9 .... 1 G G^2 G^3 G^4 G^5 G^6 G^7 G^8 G^9 ....

  and

  n H^n

  0 1 2 3 4 5 6 7 8 9 .... 1 H H^2 H^3 H^4 H^5 H^6 H^7 H^8 H^9 ....

  The values 1, G, G^2, G^3, G^4 etc get bigger and bigger.
  
  • G^0 = 1 and G^1 = G,    the initial conditions part of G^n.
  • G^n = G^(n-1) + G^(n-2), for all n greater than or equal 2    the recursive part of G^n.
  The values 1, H, H^2, H^3, H^4 etc get closer and closer to 0.
  • H^0 = 1 and H^1 = H,    the initial conditions part of H^n.
  • H^n = H^(n-1) + H^(n-2), for all n greater than or equal 2    the recursive part of H^n.
  The Fibonacci functions are related by
  • F(n)   =   G^n / sqrt(5) - H^n / sqrt(5)   =   (G^n - H^n) / sqrt(5)     which is approximately G^n / sqrt(5)   =   φ^n / sqrt(5)

• Here is a pattern involving divisibility: Each third term is even. So F(0) = 0 is even, F(3) = 2 is even, F(6) = 8 is even, F(9) = 34 is even, F(12) = 144 is even, and so on. These are the only even terms. Mathematicians often write m = 0 modulo 2 to indicate that m is even. Similarly, mathematicians often write m = 0 modulo 3 to indicate that m is divisible by 3. And so on. With this notation, the pattern we discovered, can be written as
  • F(n) = 0 modulo 2, exactly when n = 0 modulo 3
  Mathematicians can easily prove this statement.
• Once mathematicians see one pattern, they look for more of the same kind. Here is one: Each 4th term is divisible by 3. So F(0), F(4), F(8), F(12), F(16), F(20), and so on, are divisible by 3, and no other F(n) are divisible by 3. Each 5th one is divisible by 5. Each 8th one is divisible by 7. Here is a list of some of these patterns.
  • F(n) = 0 modulo 2, exactly when n = 0 modulo 3
  • F(n) = 0 modulo 3, exactly when n = 0 modulo 4
  • F(n) = 0 modulo 5, exactly when n = 0 modulo 5
  • F(n) = 0 modulo 7, exactly when n = 0 modulo 8
  • F(n) = 0 modulo 11, exactly when n = 0 modulo 10
  • F(n) = 0 modulo 13, exactly when n = 0 modulo 7
  • F(n) = 0 modulo 17, exactly when n = 0 modulo 9
  • F(n) = 0 modulo 19, exactly when n = 0 modulo 18
  • F(n) = 0 modulo 23, exactly when n = 0 modulo 24
  • F(n) = 0 modulo 29, exactly when n = 0 modulo 14
  Now we have a pattern of patterns. For each prime number p we find a number k such that
  • F(n) = 0 modulo p, exactly when n = 0 modulo k
  Mathematicians can prove these patterns.
• Mathematicians now look for patterns inside this pattern of patterns. Here is one:
  • F(3) = F(2+1) = 0 modulo 2
  • F(4) = F(3+1) = 0 modulo 3
  • F(5) = F(5+0) = 0 module 5. An exception in the pattern
  • F(8) = F(7+1) = 0 modulo 7
  • F(10) = F(11-1) = 0 modulo 11
  • F(14) = F(13+1) = 0 modulo 13
  • F(18) = F(17+1) = 0 modulo 17
  • F(18) = F(19-1) = 0 modulo 19
  • F(24) = F(23+1) = 0 modulo 23
  • F(28) = F(29-1) = 0 modulo 29
  So a pattern we see is: For all primes p with the exception of p = 5, we have that
  • Either F(p-1) = 0 modulo p, or F(p+1) = 0 modulo p
  For example, choose p = 13. Then either F(12) = 144 is divisible by 13 (it isn't) or F(14) = 377 is divisible by 13 (it is). Mathematicians can prove these patterns. Why is 5 so special? There is a connection with the occurrence of sqrt(5), also written 5^(1/2), in the formula for the golden ratio G = (1 + sqrt(5))/2.

• Whoever moves first, must take at least one penny, but not all.
• In response to the previous person's move of some pennies from the pile, the other player must take at least one penny, but at most twice as much as the other player just took.

• Some sample websites on Fibonacci numbers, collected with Phillip van Hoven:
  • Pretty simple video clip, but effective example of the fibonacci sequence (slightly vulgar).
  • May be a bit heavy on the math for this level course, but a few neat pictures for browsing.
  • Lots and lots of details on Fibonacci numbers.
  • Fibonacci and the history of the decimal system.