Leonardo Pisano (1170–1250), or Fibonacci, is perhaps best known for a remarkable sequence of numbers that arises out of a problem that involves breeding rabbits.
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:
1 
1 

3 
5 
8 

7 
9 
11 
27 

13 
15 
17 
19 
64 

21 
23 
25 
27 
29 
125 
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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: 1^{3}, 2^{3}, 3^{3}, 4^{3}, 5^{3}, and so on.
What if you did the same thing with even numbers? What happens then?
2 
2 

4 
6 
10 

8 
10 
12 
30 

14 
16 
18 
20 
68 

22 
24 
26 
28 
30 
130 
This time, the nth row has the sum n^{3} + 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 Fibonacciinspired 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:
0 
1 
1 

2 
3 
4 
5 
6 
7 
27 

8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
125 

18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
343 
“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.”
Puzzle of the Week
Sophie bought $5 worth of postage stamps of three kinds: 50cent stamps, 10cent stamps, and 1cent stamps—100 stamps in all.
How many stamps of each kind did she buy?
For the answer, go to http://www.sciencenewsforkids.org/articles/20041103/PuzzleZone.asp.