The total number of pairs, month by month, forms the sequence 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. This set of numbers is now called the Fibonacci sequence.

Fibonacci numbers come up surprisingly often in nature, from the number of petals in various flowers to the number of scales along a spiral row in a pine cone. Amazingly, the ratios of successive terms of the Fibonacci sequence get closer and closer to a specific number, often called the golden ratio.

V iswanath wondered what would happen to the Fibonacci sequence if he introduced an element of randomness.

• According to this scheme, the successive coin tosses H H T T T H, for example, would generate the sequence 1, 1, 2, 3, 1, 4, 5, 1.
• If the coin always comes up heads, for instance, the result is the original Fibonacci sequence.
• The standard Fibonacci sequence has an intriguing property.

Despite the element of chance and the resulting large fluctuations in value that characterize a random Fibonacci sequence, the absolute values of the numbers, on average, increase at a well-defined exponential rate.

Random Fibonacci sequences might have leveled off to a constant absolute value because of the subtractions, for example, but they actually escalate exponentially. In a different mathematical context involving so-called random matrix multiplication, Furstenberg and Kesten had proved that in number sequences generated by certain types of processes having an element of randomness, the value of the n th term of the sequence gets closer to the n th power of some fixed number.

Growth and decay of random Fibonacci sequences.