Lucas number: Difference between revisions
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== Definition == |
== Definition == |
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Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a [[Generalizations of Fibonacci numbers#Fibonacci integer sequences|Fibonacci integer sequence]]. The first two Lucas numbers are ''L''<sub>0</sub> = 2 and ''L''<sub>1</sub> = 1 as opposed to the first two Fibonacci numbers ''F''<sub>0</sub> = 0 and ''F''<sub>1</sub> = 1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties. |
Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a [[Generalizations of Fibonacci numbers#Fibonacci integer sequences|Fibonacci integer sequence]]. The first two Lucas numbers are ''L''<sub>0</sub> = 2 and ''L''<sub>1</sub> = 1 as opposed to the first two Fibonacci numbers ''F''<sub>0</sub> = 0 and ''F''<sub>1</sub> = 1.<ref name = "NKS note a">''[[A New Kind of Science]]'' [https://wolframscience.com/nks/notes-3-5--lucas-numbers/]</ref> Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties. |
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The Lucas numbers may thus be defined as follows: |
The Lucas numbers may thus be defined as follows: |
Revision as of 15:43, 3 September 2020
This article includes a list of general references, but it lacks sufficient corresponding inline citations. (December 2019) |
The Lucas numbers or Lucas series are an
The Lucas sequence has the same recursive relationship as the Fibonacci sequence, where each term is the sum of the two previous terms, but with different starting values.[1] This produces a sequence where the ratios of successive terms approach the golden ratio, and in fact the terms themselves are roundings of integer powers of the golden ratio.[2] The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between.[3]
The first few Lucas numbers are
- 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ....
Definition
Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are L0 = 2 and L1 = 1 as opposed to the first two Fibonacci numbers F0 = 0 and F1 = 1.[4] Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.
The Lucas numbers may thus be defined as follows:
(where n belongs to the natural numbers)
The sequence of the first twelve Lucas numbers is:
All Fibonacci-like integer sequences appear in shifted form as a row of the Wythoff array; the Fibonacci sequence itself is the first row and the Lucas sequence is the second row. Also like all Fibonacci-like integer sequences, the ratio between two consecutive Lucas numbers converges to the golden ratio.
Extension to negative integers
Using Ln−2 = Ln − Ln−1, one can extend the Lucas numbers to negative integers to obtain a doubly infinite sequence:
- ..., −11, 7, −4, 3, −1, 2, 1, 3, 4, 7, 11, ... (terms for are shown).
The formula for terms with negative indices in this sequence is
Relationship to Fibonacci numbers
The Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following:
- , and thus as approaches +∞, the ratio approaches
- ; in particular,
Their closed formula is given as:
where is the golden ratio. Alternatively, as for the magnitude of the term is less than 1/2, is the closest integer to or, equivalently, the integer part of , also written as .
Combining the above with
a formula for is obtained:
Congruence relations
If Fn ≥ 5 is a Fibonacci number then no Lucas number is divisible by Fn.
Ln is congruent to 1 mod n if n is prime, but some composite values of n also have this property. These are the Fibonacci pseudoprimes.
Ln - Ln-4 is congruent to 0 mod 5.
Lucas primes
A Lucas prime is a Lucas number that is prime. The first few Lucas primes are
- 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, ... (sequence A005479 in the OEIS).
The indices of these primes are (for example, L4 = 7)
- 0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, ... (sequence A001606 in the OEIS).
If Ln is prime then n is 0, prime, or a power of 2.[5] L2m is prime for m = 1, 2, 3, and 4 and no other known values of m.
Generating series
Let
be the
which can be rearranged as
The partial fraction decomposition is given by
where is the golden ratio and is its conjugate.
Lucas polynomials
In the same way as
See also
References
- ^ Weisstein, Eric W. "Lucas Number". mathworld.wolfram.com. Retrieved 2020-08-11.
- ISBN 978-0-374-53563-6.
- ISBN 978-0-374-53563-6.
- ^ A New Kind of Science [1]
- Prime Pages.
External links
- "Lucas polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- Weisstein, Eric W. "Lucas Number". MathWorld.
- Weisstein, Eric W. "Lucas Polynomial". MathWorld.
- "The Lucas Numbers", Dr Ron Knott
- Lucas numbers and the Golden Section
- A Lucas Number Calculator can be found here.
- OEIS sequence A000032 (Lucas numbers beginning at 2)