Interesting Article on Fibonacci

[PAB]

Hi Everyone,

I Know that there has been an Interest about Fibonacci and his Work on this Forum in the Past, so I thought this might be of Interest.

The Life and Numbers of Fibonacci by R.Knott, D.A.Quinney and PASS Maths :-
http://plus.maths.org/issue3/fibonacci/index.html

I Found the Following Links ( Originally Posted by thornc in September Last Year ) from a Search of this Forum and thought they Might Also be of Interest.

Fibonacci :-
http://mathworld.wolfram.com/FibonacciNumber.html

Lucas Numbers ( the Companions to the Fibonacci Numbers ) :-
http://mathworld.wolfram.com/LucasNumber.html

The Golden Number :-
http://goldennumber.net/

All the Best.
PAB

 

[Karnac]

Another article on the topic.....

Fibonacci's Other Numbers
Ivars Peterson

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


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?

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 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:

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."

 

[johnph77]

more grist for the math-hungry.....

In addition to the Fibonacci numbers, there are also the Lucas numbers, which are calculated the same way as the Fibonacci sequence except that the starting two numbers are 2,1. This makes the start of the sequence:

2, 1, 3, 4, 7, 11, 18, 29, 47, 76.....

There's also a similar sequence utilizing three numbers (the name doesn't come immediately to mind) that starts:

1, 1, 1, 3, 5, 9, 17, 31, 57, 95.....

This sequence will produce only odd numbers. But starting with 1,2,3:

1, 2, 3, 6, 11, 20, 37, 68.....

(odd, even, odd, even.....)

But the Fibonacci sequence occurs more often in nature than any other sequence and is the integer expression closest to what is known as the Golden Ratio, equivalent to about 1::1.618. This ratio also appears extensively in the art world (see Mondrian's abstracts, for example) and is the ratio widely believed to be the most pleasant and receptive to the human eye, whether height to width or width to height.

 

[GillesD]

Fibonacci and Lucas numbers

Fibonacci numbers

For a 6/49 lottery, there are 8 Fibonacci numbers (1, 2, 3, 5, 8, 13, 21 and 34) or 16.33%.

So after 2205 draws, we should expect that percentage of the winning numbers to be Fibonacci numbers. This amounts to 2,160 numbers, if the bonus number is not considered. Actually, the values are 2,126 Fibonacci numbers and 11,104 other numbers. The difference between theory and reality is a big 0.26%.

We can also look at one other characteristic: the ratio of Fibonacci / other numbers in a draw. This can go from 0/6 to 6/0. These are the ratios obtained after 2205 draws. The data is listed in the following order: the ratio Fibonacci / other numbers, the actual number of time this ratio occurred, the theorical number of times it should have occurred and these two same values but in percentages.

0 / 6 : 734 -- 709 -- 33.3% -- 32.2%
1 / 5 : 880 -- 945 -- 39.9% -- 42.9%
2 / 4 : 444 -- 448 -- 20.1% -- 20.3%
3 / 3 : 110 -- 094 -- 5.0% -- 4.3%
4 / 2 : 007 -- 009 -- 0.3% -- 0.4%
5 / 1 : 000 -- 000 -- 0.0% -- 0.0%
6 / 0 : 000 -- 000 -- 0.0% -- 0.0%


Lucas numbers

For a 6/49 lottery, there are also 8 Lucas numbers (1, 3, 4, 7, 11, 18, 29 and 47) or 16.33%.

So after 2205 draws, we should expect that percentage of the winning numbers to be Lucas numbers. This amounts to 2,160 numbers, if the bonus number is not considered. Actually, the values are 2,143 Lucas numbers and 11,087 other numbers. The difference between theory and reality is a big 0.13%.

We can also look at one other characteristic: the ratio of Lucas / other numbers in a draw. This can go from 0/6 to 6/0. These are the ratios obtained after 2205 draws. The data is listed in the following order: the ratio Lucas / other numbers, the actual number of time this ratio occurred, the theorical number of times it should have occurred and these two same values but in percentages.

0 / 6 : 694 -- 709 -- 31.5% -- 32.2%
1 / 5 : 936 -- 945 -- 42.4% -- 42.9%
2 / 4 : 441 -- 448 -- 20.0% -- 20.3%
3 / 3 : 092 -- 094 -- 4.2% -- 4.3%
4 / 2 : 011 -- 009 -- 0.5% -- 0.4%
5 / 1 : 001 -- 000 -- 0.0% -- 0.0%
6 / 0 : 000 -- 000 -- 0.0% -- 0.0%

 

[PAB]

Hi,

johnph77,
The Sequence of Numbers Utilizing Three Numbers that you could Not Recollect is Called the Tribonacci Series ( see Below ).

Here are a Few Other Series of Numbers that Might be of Interest.

The Fibonacci Series is 0,1,1,2,3,5,8,13,21,34,55,89 … :-
http://mathworld.wolfram.com/FibonacciNumber.html
( Each is the Sum of the Two Preceding Numbers in the Sequence ).
The Ratio is 1.618033988 … ( The Golden Ratio [ Mean ] )

The Tribonacci Series :-
http://mathworld.wolfram.com/TribonacciNumber.html
( Each is the Sum of the Three Preceding Numbers in the Sequence ).

The Tetranacci Series :-
http://mathworld.wolfram.com/TetranacciNumber.html
( Each is the Sum of the Four Preceding Numbers in the Sequence ).

The Pentanacci Series :-
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001591
( Each is the Sum of the Five Preceding Numbers in the Sequence ).

The Hexanacci Series is :-
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A001592
( Each is the Sum of the Six Preceding Numbers in the Sequence ).

The Heptanacci Series is :-
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A066178
( Each is the Sum of the Seven Preceding Numbers in the Sequence ).

All the Best.
PAB