Lucas
Series - a code to find n
elements
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The
French mathematician, Edouard Lucas (19th century), found a sequence of
numbers (named the Lucas
series) similar to the sequence found in the Fibonacci numbers.
The
Fibonacci rule of adding the latest two integers
to obtain the next number is kept, but here we start from 2 and 1,
instead of 0
and 1 used for the Fibonacci series. |
The
series, also 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 > 2
L1 = 2
L2 = 1
Like the Fibonacci series,
each Lucas number
is defined to be the sum of its two immediate previous terms. The ratio
between
two consecutive Lucas numbers converges to the golden ratio.
However,
the first
two Lucas numbers are L1
=
2 and L2 = 1
(instead of Fibonacci's F1
= 0 and F2 =
1), and the properties
of Lucas numbers are therefore different from those of Fibonacci
numbers.
We can generate a fast code
to obtain the
Lucas series (n is the input
parameter and it’s the number of terms that we’ll calculate for the
series).
function L =
lucas_series(n)
%
Define the first two elements
L(1) =
2;
L(2) =
1;
%
Calculate as many numbers as needed
for i = 3 :
n
L(i)
= L(i-1) + L(i-2);
end
Let’s test that function:
clear,
clc, format long
L =
lucas_series(15)
%
Let's find the ratio of consecutive
%
elements in the series
for i = 1 :
length(L)-1
list
= [L(i) L(i+1) L(i+1)/L(i)]
end
And we get the list of 15
terms and the ratio of consecutive numbers:
L = 2
1
3
4
7
11
18
29
47
76
123
199 322 521
843
list = 2.00000000000000 1.00000000000000 0.50000000000000
list = 1
3
3
list = 3.00000000000000 4.00000000000000 1.33333333333333
list = 4.00000000000000 7.00000000000000 1.75000000000000
list = 7.00000000000000 11.00000000000000 1.57142857142857
list = 11.00000000000000 18.00000000000000 1.63636363636364
list = 18.00000000000000 29.00000000000000 1.61111111111111
list = 29.00000000000000 47.00000000000000 1.62068965517241
list = 47.00000000000000 76.00000000000000 1.61702127659574
list = 1.0e+002 *
0.76000000000000 1.23000000000000 0.01618421052632
list = 1.0e+002 *
1.23000000000000 1.99000000000000 0.01617886178862
list = 1.0e+002 *
1.99000000000000 3.22000000000000 0.01618090452261
list = 1.0e+002 *
3.22000000000000 5.21000000000000 0.01618012422360
list = 1.0e+002 *
5.21000000000000 8.43000000000000 0.01618042226488
We can see that, at the end, we end with a
ratio that is an approximation to the so called golden ratio
(1.6180...).
There are some interesting
facts about the
Lucas series and the Fibonacci numbers. You can look them here (opens new window).
From
'Lucas Series' to home
From
'Lucas
Series' to Calculus Problems
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