In the familiar Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, 21,... the next number is always the sum of the previous two (0+1=1, 1+1=2, 1+2=3, 2 +3=5, 3+5=8,...). The first two numbers are given as 0 and 1.

Suppose we make a sequence where the next number is always the sum of the previous three numbers. We specify the first three numbers to be 0, 0, and 1. So the sequence is 0, 0, 1, 1, 2, 4, 7, 13, 24, 37, 61,... . What can you say about this sequence?

  Getting Started

1. How big, approximately, is the n th number in this sequence? Does it grow faster than the Fibonacci sequence?
   
2. For the Fibonacci sequence the ratios of successive numbers approach the golden ratio, that is, 1/2, 2/3, 3/5, 5/8, ... get closer and closer to (1+\sqrt{5})/2. (Can you say why this happens?) Do the ratios of successive numbers in this sequence approach any particular number? What is it?

  Going Deeper

Where might this new sequence of numbers appear naturally? (Compare to applications of the Fibonacci sequence.)

Communicated by Eileen Herlihy.