Ivars Peterson's MathTrek
The problem is contained in the third section of Fibonacci's 1202 book Liber abaci:
A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive. The numbers in the resulting sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . ., are now known as Fibonacci numbers. Each number is the sum of the two preceding numbers.
Fibonacci's writings also include a wide variety of astute observations on numbers patterns and important results in number theory.
Here's an interesting pattern that Fibonacci explored involving a triangle of odd whole numbers:
Using only odd numbers, place the first one in the first row, the second two in the second row, the next three in the third row, the next four in the fourth row, and so on. Then find the sum of the numbers in each row. The resulting sums are 1, 8, 27, 64, 125, and so on. You get a sequence of consecutive cubes: 13, 23, 33, 43, 53, and so on.
What if you did the same thing with even numbers? What happens then?
This time, the nth row has the sum n3 + n.
Richard L. Ollerton of the University of Western Sydney and Anthony G. Shannon of Warrane College, University of New South Wales, Australia, explore many such Fibonacci-inspired number arrays in the current issue of the Journal of Recreational Mathematics.
They suggest a variety of ways in which interested readers can explore such arrays further. What happens when different sequences are used? What happens if row lengths go up in larger steps?
Here's an example with odd integer cube row sums:
"Generalized Fibonacci arrays have attractive properties and could provide a wealth of further activities for exploration," Ollerton and Shannon write. "We have considered arithmetic progressions but geometric or other sequences whose partial sums are known, together with a wider variety of row length sequences, could also be studied."
Copyright © 2005 by Ivars Peterson
Ollerton, R.L., and A.G. Shannon. 2004. Fibonacci's odd number array. Journal of Recreational Mathematics 32(No. 3):198-204.
Peterson, I. 2004. Counting on Fibonacci. MAA Online (May 3).
For additional information about Fibonacci formulae and number patterns, see http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibFormulae.html.
Biographical information about Fibonacci is available at http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Fibonacci.html.