Rankine–Hugoniot conditions

2013-10-19T14:12:35Z

87.114.124.81: /* Hugoniot elastic limit */

{{distinguish2|[[Lucas sequence]]s, which are a generic class of sequences to which the Lucas numbers belong}}

The '''Lucas numbers''' or '''Lucas series''' are an [[integer sequence]] named after the mathematician [[Édouard Lucas|François Édouard Anatole Lucas]] (1842–1891), who studied both that sequence and the closely related [[Fibonacci number]]s. Lucas numbers and Fibonacci numbers form complementary instances of [[Lucas sequence]]s.

== Definition ==
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''0 = 2 and ''L''1 = 1 as opposed to the first two Fibonacci numbers ''F''0 = 0 and ''F''1 = 1. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.

The Lucas numbers may thus be defined as follows:
:<math>
L_n :=
\begin{cases}
2 & \text{if } n = 0; \\
1 & \text{if } n = 1; \\
L_{n-1}+L_{n-2} & \text{if } n > 1. \\
\end{cases}
</math>

The sequence of Lucas numbers is:
:<math>2,\;1,\;3,\;4,\;7,\;11,\;18,\;29,\;47,\;76,\;123,\; \ldots\;</math>{{OEIS|id=A000032}}.

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 [[limit of a sequence|converges]] to the [[golden ratio]].

==Extension to negative integers==
Using ''L''''n''−2 = ''L''''n'' − ''L''''n''−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 <math>L_n</math> for <math>-5\leq{}n\leq5</math> are shown).
The formula for terms with negative indices in this sequence is
: <math>L_{-n}=(-1)^nL_n.\!</math>

==Relationship to Fibonacci numbers==
The Lucas numbers are related to the Fibonacci numbers by the identities
* <math>\,L_n = F_{n-1}+F_{n+1}=F_n+2F_{n-1}</math>
* <math>\,L_{m+n} = L_{m+1}F_{n}+L_mF_{n-1}</math>
* <math>\,L_n^2 = 5 F_n^2 + 4 (-1)^n</math>, and thus as <math>n\,</math> approaches [[Extended real number line|+∞]], the ratio <math>\frac{L_n}{F_n}</math> approaches <math>\sqrt{5}.</math>
* <math>\,F_{2n} = L_n F_n</math>
* <math>\,F_{n+k} + (-1)^k F_{n-k} = L_k F_n</math>
* <math>\,F_n = {L_{n-1}+L_{n+1} \over 5}</math>

Their [[Closed-form expression|closed formula]] is given as:
:<math>L_n = \varphi^n + (1-\varphi)^{n} = \varphi^n + (- \varphi)^{- n}=\left({ 1+ \sqrt{5} \over 2}\right)^n + \left({ 1- \sqrt{5} \over 2}\right)^n\, ,</math>

where <math>\varphi</math> is the [[Golden ratio]]. Alternatively, as for <math>n>1</math> the magnitude of the term <math>(-\varphi)^{-n}</math> is less than 1/2, <math>L_n</math> is the closest integer to <math>\varphi^n</math> or, equivalently, the integer part of <math>\varphi^n+1/2</math>, also written as <math>\lfloor \varphi^n+1/2 \rfloor</math>.

Conversely, <math>\varphi^n = {{L_n + F_n \sqrt{5}} \over 2}</math>.

==Congruence relations==
If ''F''''n'' ≥ 5 is a Fibonacci number then no Lucas number is divisible by ''F''''n''.

''L''''n'' is congruent to 1 mod ''n'' if ''n'' is prime, but some composite values of ''n'' also have this property.

== Lucas primes ==
A '''Lucas prime''' is a Lucas number that is [[prime number|prime]]. The first few Lucas primes are
: 2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, ... {{OEIS|id=A005479}}.
If ''Ln'' is prime then ''n'' is either 0, prime, or a power of 2.<ref>Chris Caldwell, "[http://primes.utm.edu/glossary/page.php?sort=LucasPrime The Prime Glossary: Lucas prime]" from The [[Prime Pages]].</ref> ''L''2''m'' is prime for ''m'' = 1, 2, 3, and 4 and no other known values of ''m''.

==Lucas polynomials==
In the same way as [[Fibonacci polynomial]]s are derived from the [[Fibonacci number]]s, the '''Lucas polynomials''' ''L''''n''(''x'') are a [[polynomial sequence]] derived from the Lucas numbers

==See also==
*[[Fibonacci prime]]

== References ==
{{reflist}}

==External links==
*{{MathWorld|urlname=LucasNumber|title=Lucas Number}}
*{{MathWorld | urlname=LucasPolynomial | title=Lucas Polynomial}}
*[http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/lucasNbs.html Dr Ron Knott]
*[http://milan.milanovic.org/math/english/lucas/lucas.html Lucas numbers and the Golden Section]
*[http://www.plenilune.pwp.blueyonder.co.uk/fibonacci-calculator.asp A Lucas Number Calculator can be found here.]
*[http://nakedprogrammer.com/LucasNumbers.aspx A Tutorial on Generalized Lucas Numbers]
*[http://oeis.org/A000204 Lucas Numbers] from [http://oeis.org/ The On-Line Encyclopedia of Integer Sequences].

[[Category:Integer sequences]]
[[Category:Fibonacci numbers]]

[[bn:লুকাস ধারা]]
[[de:Lucas-Folge]]
[[fr:Suite de Lucas]]
[[he:סדרת לוקאס]]
[[nl:Rij van Lucas]]
[[pt:Sequência de Lucas]]

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Rankine–Hugoniot conditions