^^ Richard K Guy in The Second Strong Law of Small Numbers in The Mathematics Magazine (1990), Vol 63, pages 3-20

mentions the Pennies Puzzle 1 and Pennies Puzzle 2 on the Fibonacci Puzzle page and that only one of them is truly Fibonacci.

This fun paper also has several other Fibonacci Forgeries including ones on partitions of n, rooted trees with one label, the number of disconnected graphs on n+1 vertices and the number of connected graphs on n+2 vertices which have just one cycle.

There are many other forgeries in the paper to do with primes, Catalan numbers, binomial and trinomial numbers, mixing some genuine examples with the forgeries. His whole point is that There are not enough small numbers to meet the many demands made of them and so we are bound to be fooled with small examples of a problem if we are not careful!

Fibonacci Home PageF!

The Fibonacci Numbers in

Fibonacci - the man and His formulae for Pi

Times

The next topics...

Fibonacci, Phi and Lucas numbers Formulae

Links and References

^ The Lucas Numbers

© 1996-1999 Dr Ron Knott [email protected] 29 August 1999

We have seen in earlier pages that there is another series quite similar to the Fibonacci series that often occurs when working with the Fibonacci series. Edouard Lucas (1842-1891) (who gave the name "Fibonacci Numbers" to the series written about by Leonardo of Pisa) studied this second series of numbers: 2, 1, 3, 4, 7, 11, 18, .. called the Lucas numbers in his honour. On this page we examine some of the interesting properties of the Lucas numbers themselves as well as looking at its close relationship with the Fibonacci numbers.

Contents

The i m line means there is a Things to do investigation at the end of the section.

M Other starting values for a "Fibonacci" seriesi m m The Lucas series

M Two formulae relating the Lucas and Fibonacci numbers i m M A formula for the Lucas Numbers involving Phi and phi i h * A number trick based on Phi, Lucas and Fibonacci numbers! J An even more complicated-looking variation! J Why does it work? M The Lucas Numbers in Pascal's Triangle References

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

Other starting values for a "Fibonacci" series

The definition of the Fibonacci series is:

Fn+1 = Fn-1 + |
Fn , if n>1 |

Fo = |
0 |

Fl = |
1 |

What if we have the same general rule: add the latest two values to get the next but we started with different values instead of 0 and 1?

1. The Fibonacci series starts with 0 and 1. What if we started a

"Fibonacci" series with 1 and 2, using the same general rule is for the

Fibonacci series proper, so that F0 = 1 and F1 = 2? What numbers follow?

2. What if we started with 2 and 3 so that F0 = 2 and F1 = 3?

3. What other starting values give the same series as the previous two questions?

4. The simplest values to start with are

all of which we recognise as (part of) the Fibonacci series after a few terms.

The next two simplest numbers are 2 and 1.

What if we started with 2 and 1 so that F0 = 2 and F1 = 1? Does this become part of the Fibonacci series too?

6. Investigate what happens to the ratio of successive terms in the series of the earlier questions. We know that for the Fibonacci series, the ratio gets closer and closer to Phi = (V5+l)/2. Does it look as (oh dear, I feel a pun coming on: Lucas ©) if all the series, no matter what starting values we choose, eventually have successive terms whose ratio is Phi?

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

The French mathematician, Edouard Lucas (1842-1891), who gave the series of numbers 0, 1, 1, 2, 3, 5, 8, 13, the name the Fibonacci Numbers. found another similar series: 2, 1, 3, 4, 7, 11, 18, ... . The Fibonacci rule of adding the latest two to get the next is kept, but here we begin with 2 and 1 (in this order).

The series, called the Lucas Numbers after him, is defined as follows: where we write its members as Ln, for

Lucas:

Ln - Ln-1 + |
Ln-2 for n>1 |

Lo |
- 2 |

L1 |
- 1 |

and here are some more values of Ln together with the Fibonacci numbers for comparison:

n: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
... |

Fn: |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 | |

Ln: |
2 |
1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
76 |
123 |

The Lucas numbers have lots of properties similar to those of Fibonacci numbers and, uniquely among the series you examined in the Things To Do section above, the Lucas numbers often occur in various formulae for the Fibonacci Numbers. Also, if you look at many formulae for the Lucas numbers, you will find the Fibonacci series is there too. The next section introduces you to some of these equations. So of all the 'general Fibonacci' series, these two seem to be the most important.

For instance, here is the graph of the ratios of successive Lucas numbers:

7 11 18

18=1611L

In fact, for every series formed by adding the latest two values to get the next, and no matter what two values we start with we will always end up having terms whose ratio is Phi=16180339.. eventually!

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

Two formulae relating the Lucas and Fibonacci numbers

Suppose we add up alternate Fibonacci numbers, Fn_1 + Fn+1 that is, what do you notice about the two Fibonacci numbers either side of a Lucas number in the table below: eg

n: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
... |

Fn: |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 | |

Ln: |
2 |
1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
76 |
123 |

Now try your guess on some other Lucas numbers.

This gives our first equation connecting the Fibonacci numbers F(n) to the Lucas numbers L(n):

What about adding alternate Lucas numbers?

Fn: |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 | |

Ln: |
2 |
1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
76 |
123 |

The sum of L(2)=3 and L(4)=7 is not F(3)=2 However, try a few more additions in this pattern:

L(1)=1 and L(3)= 4 so their sum is 5 whereas F(2)=1;

L(2)=3 and L(4)= 7 so their sum is 10 whereas F(3)=2;

L(3)=4 and L(5)=11 so their sum is 15 whereas F(4)=3;

L(4)=7 and L(6)=18 so their sum is 25 whereas F(5)=5;

Have you spotted the pattern?

5 F(n) = L(n-1) + L(n+1) for all integers n a. What about the Fibonacci numbers that are TWO places away from Lucas(n)?

n: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
... |

Fn: |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 | |

Ln: |
2 |
1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
76 |
123 |

What is the relationship between F(n-2), and F(n+2) that will give L(n)?

b. There is also a relationship between F(n-3) and F(n+3) that gives L(n).

n: |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
... |

Fn: |
0 |
1 |
1 |
2 |
3 |
5 |
8 |
13 |
21 |
34 |
55 | |

Ln: |
2 |
1 |
3 |
4 |
7 |
11 |
18 |
29 |
47 |
76 |
123 |

What is it? Write it down as a mathematical formula.

d. Look back at the formula you have just found. Do they work if n is negative (n<0)?

e. Can you write down a general expression that relates F(n-k) and F(n+k) to give L(n)?

2. How about the other way round now!

a. We have already found the relationship between L(n-1) and L(n+1) that gives F(n) - in fact 5F(n) - above.

b. And now try using L(n-3) and L(n+3) to get F(n).

c. .. and how can you use L(n-4) and L(n+4) to derive F(n)?

d. Look back at the formula you have just found. Do they work if n is negative (n<0)?

e. Can you write down a general expression that relates L(n-k) and

L(n+k) to give F(n)? 3. Now - the really interesting part!

Have you spotted a pattern in these patterns?

If you have, can you write down a mathematical expression which covers ALL the formula found in this Things To Do section?

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843 ..More..

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