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A Sequence You Will Want to Share

You have probably seen this sequence, which has been passed around alot:

1, 11, 21, 1211, 111221, 312211, ?

Question 1: What are the next three numbers in the sequence?

Question 2: If it ever occurred, the only sequence of numbers that would be unchanged is "22"...why?

Question 3: Explain why one of the first two digits, starting from the root "1" as shown, will always be a 1.

Question 4: Show that the first digit cannot be a 1 for more than two iterations.

Note 1: The rate of growth for this sequence is exponential...and it has been proven that its "asymptotic growth factor is algebraic of degree 71."

Note 2: Believe or not, it has been proven that this sequence shares properties similar to chemical macromolecules such as RNA, DNA, and double/multiple helixes.


Hint: It is called the "look and say sequence"...so apply that name when you study the sequence...and be a "counter"!


Solution Commentary: For more information on this sequence, consider either the Mathworld web site or the Wikipedia web site.