(We regularly translate mathematical articles and various others and add them to the Bengali Wikipedia. As part of this, today we are presenting the newly added article on the Lucas sequence. This article is freely available under the GNU Free Documentation License on the Bengali Wikipedia.)
From Wikipedia, the free encyclopedia
Lucas sequence is an integer sequence discovered by François Édouard Anatole Lucas. Fibonacci sequence and Lucas sequence are both examples of the Lucas sequences.
Each number in the Lucas sequence is equal to the sum of the previous two numbers. Therefore, the ratio of two consecutive terms of the Lucas sequence equals the golden ratio.
The difference between the Fibonacci and Lucas sequences is that the first two numbers of the Lucas sequence are L0 = 2 and L1 = 1 (in the Fibonacci sequence, these two numbers are 0 and 1).
Thus, the definition of the Lucas sequence is:
So, the numbers in the Lucas sequence are:
- 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, and so on.
1. Negative Sequence
To extend the Lucas sequence to negative indices, the formula Ln-2 = Ln – Ln-1 can be used. As a result, the sequence of negative-indexed Lucas numbers is:
… -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, …
2. Relation to the Fibonacci sequence
The relationship between the Lucas and Fibonacci sequences can be expressed by the following formulas:
And the formula for the nth term of the Lucas sequence is:
where 
is the golden ratio.
Additionally:
As 
approaches infinity, the value of 
approaches 
.
3. Congruence formula
Ln is congruent to 1 mod n if n is a prime number. However, this feature can also be seen for many composite values of n.
4. Lucas Prime Numbers
Lucas primes are those numbers in the Lucas sequence that are also prime numbers. The first few Lucas primes are:
2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, …
Except for n = 0, 4, 8, 16, if Ln is a prime number, then n is also a prime number. The converse is not necessarily true.
5. External links
- Lucas Numbers at MathWorld
- Webpage by Dr. Ron Knott
- Lucas sequence and the golden ratio
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golden ratio somporke bangla lekha aase ki?
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