A Special Sequence

1. A Consequential Countdown

A Special Sequence

Remember the puzzling countdown (see "A Consequential Countdown"): 55, 34, 21, 13, 8, 5, 3, 2, 1, 1?

Look at it in reverse order: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55. Do you see a pattern? Can you predict what number would come next?

The numbers belong to a famous sequence named for the Italian mathematician Fibonacci, who lived more than 700 years ago.

Leonardo of Pisa, also known as Fibonacci. He lived from about 1170 to 1240.

Each consecutive number is the sum of the two numbers that precede it. Thus, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, and so on.

The ninth number of the Fibonacci sequence is 34 and the tenth is 55, so the eleventh is 89. What is the twelfth Fibonacci number? What is the sixteenth?

TRY IT!

Here are some additional number sequences. See if you can fill in the missing numbers and figure out the rules for making each sequence.

1, 3, 4, 7, 11, 18, 29, 47, ____, 122, …

3, 6, 12, 24, 48, ____, 192, 384, …

1, 3, 6, 10, 15, 21, 28, 36,____, 55, 66, 78, …

1, 4, 9, 16, ____, 36, 49, 64, 81, 100, …

Number sequences have intrigued mathematicians for centuries. The first example given above gives the Lucas numbers, honoring the French mathematician Édouard Lucas, who studied the Fibonacci sequence.

Lucas worked out what would happen if you started with any two whole numbers, then followed the Fibonacci rule. He discovered many interesting new sequences and number patterns.

Mathematician Neil Sloane has been collecting number sequences ever since he was a student at Cornell University in the 1960s. He described nearly six thousand examples in his book the Encyclopedia of Integer Sequences. The current online version (The On-Line Encyclopedia of Integer Sequences) catalogs more than 300,000 sequences.

Mathematicians and other researchers use his book and database as a reference for counting or tabulating things that involve number sequences, from the number of atoms in various molecules to different types of knots.

See also "Deciphering Integer Sequences" and "Sequence Puzzles."

Answers:

The twelfth Fibonacci number is 144. The sixteenth is 987.

The missing numbers are: 76, 96, 44, and 25.

The first set of numbers is an example of a Lucas sequence that begins with 1 and 3, then 1 + 3 = 4, 3 + 4 = 7, 4 + 7 = 11, 7 + 11 = 18, and so on.

In the second sequence, the numbers double again and again.

In the third, you start with 1, add 2 to get 3, then add 3 to get 6, then add 4 to the new answer to get 10, and so on.

The fourth sequence consists of consecutive perfect squares: 1 ✕ 1 = 1, 2 ✕ 2 = 4, 3 ✕ 3 = 9, and so on.

NEXT: Nature's Numbers