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- Lucas number
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- Lucas number - Wikipedia
- The First 200 Lucas numbers and their factors - University of Surrey
- Lucas Numbers - GeeksforGeeks
- Lucas Number -- from Wolfram MathWorld
- Lucas Numbers | Brilliant Math & Science Wiki
- Lucas Sequence Calculator - Check Lucas Numbers & Calculate …
- Golden Ratio, Fibonacci Numbers and Lucas Numbers
- Introduction to the Fibonacci and Lucas numbers - Wolfram
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- A000032 - OEIS
Lucas number GudangMovies21 Rebahinxxi LK21
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences.
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. 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. 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.
The first few Lucas numbers are
2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... . (sequence A000032 in the OEIS)
which coincides for example with the number of independent vertex sets for cyclic graphs
C
n
{\displaystyle C_{n}}
of length
n
≥
2
{\displaystyle n\geq 2}
.
Definition
As with the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediately previous terms, thereby forming a Fibonacci integer sequence. The first two Lucas numbers are
L
0
=
2
{\displaystyle L_{0}=2}
and
L
1
=
1
{\displaystyle L_{1}=1}
, which differs from the first two Fibonacci numbers
F
0
=
0
{\displaystyle F_{0}=0}
and
F
1
=
1
{\displaystyle F_{1}=1}
. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.
The Lucas numbers may thus be defined as follows:
L
n
:=
{
2
if
n
=
0
;
1
if
n
=
1
;
L
n
−
1
+
L
n
−
2
if
n
>
1.
{\displaystyle 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}}}
(where n belongs to the natural numbers)
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
L
n
−
2
=
L
n
−
L
n
−
1
{\displaystyle 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
L
n
{\displaystyle L_{n}}
for
−
5
≤
n
≤
5
{\displaystyle -5\leq {}n\leq 5}
are shown).
The formula for terms with negative indices in this sequence is
L
−
n
=
(
−
1
)
n
L
n
.
{\displaystyle L_{-n}=(-1)^{n}L_{n}.\!}
Relationship to Fibonacci numbers
The Lucas numbers are related to the Fibonacci numbers by many identities. Among these are the following:
L
n
=
F
n
−
1
+
F
n
+
1
=
2
F
n
+
1
−
F
n
{\displaystyle L_{n}=F_{n-1}+F_{n+1}=2F_{n+1}-F_{n}}
L
m
+
n
=
L
m
+
1
F
n
+
L
m
F
n
−
1
{\displaystyle L_{m+n}=L_{m+1}F_{n}+L_{m}F_{n-1}}
F
2
n
=
L
n
F
n
{\displaystyle F_{2n}=L_{n}F_{n}}
F
n
+
k
+
(
−
1
)
k
F
n
−
k
=
L
k
F
n
{\displaystyle F_{n+k}+(-1)^{k}F_{n-k}=L_{k}F_{n}}
2
F
2
n
+
k
=
L
n
F
n
+
k
+
L
n
+
k
F
n
{\displaystyle 2F_{2n+k}=L_{n}F_{n+k}+L_{n+k}F_{n}}
L
2
n
=
5
F
n
2
+
2
(
−
1
)
n
=
L
n
2
−
2
(
−
1
)
n
{\displaystyle L_{2n}=5F_{n}^{2}+2(-1)^{n}=L_{n}^{2}-2(-1)^{n}}
, so
lim
n
→
∞
L
n
F
n
=
5
{\displaystyle \lim _{n\to \infty }{\frac {L_{n}}{F_{n}}}={\sqrt {5}}}
.
|
L
n
−
5
F
n
|
=
2
φ
n
→
0
{\displaystyle \vert L_{n}-{\sqrt {5}}F_{n}\vert ={\frac {2}{\varphi ^{n}}}\to 0}
L
n
+
k
−
(
−
1
)
k
L
n
−
k
=
5
F
n
F
k
{\displaystyle L_{n+k}-(-1)^{k}L_{n-k}=5F_{n}F_{k}}
; in particular,
F
n
=
L
n
−
1
+
L
n
+
1
5
{\displaystyle F_{n}={L_{n-1}+L_{n+1} \over 5}}
, so
5
F
n
+
L
n
=
2
L
n
+
1
{\displaystyle 5F_{n}+L_{n}=2L_{n+1}}
.
Their closed formula is given as:
L
n
=
φ
n
+
(
1
−
φ
)
n
=
φ
n
+
(
−
φ
)
−
n
=
(
1
+
5
2
)
n
+
(
1
−
5
2
)
n
,
{\displaystyle 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}\,,}
where
φ
{\displaystyle \varphi }
is the golden ratio. Alternatively, as for
n
>
1
{\displaystyle n>1}
the magnitude of the term
(
−
φ
)
−
n
{\displaystyle (-\varphi )^{-n}}
is less than 1/2,
L
n
{\displaystyle L_{n}}
is the closest integer to
φ
n
{\displaystyle \varphi ^{n}}
or, equivalently, the integer part of
φ
n
+
1
/
2
{\displaystyle \varphi ^{n}+1/2}
, also written as
⌊
φ
n
+
1
/
2
⌋
{\displaystyle \lfloor \varphi ^{n}+1/2\rfloor }
.
Combining the above with Binet's formula,
F
n
=
φ
n
−
(
1
−
φ
)
n
5
,
{\displaystyle F_{n}={\frac {\varphi ^{n}-(1-\varphi )^{n}}{\sqrt {5}}}\,,}
a formula for
φ
n
{\displaystyle \varphi ^{n}}
is obtained:
φ
n
=
L
n
+
F
n
5
2
.
{\displaystyle \varphi ^{n}={{L_{n}+F_{n}{\sqrt {5}}} \over 2}\,.}
For integers n ≥ 2, we also get:
φ
n
=
L
n
−
(
−
φ
)
−
n
=
L
n
−
(
−
1
)
n
L
n
−
1
−
L
n
−
3
+
R
{\displaystyle \varphi ^{n}=L_{n}-(-\varphi )^{-n}=L_{n}-(-1)^{n}L_{n}^{-1}-L_{n}^{-3}+R}
with remainder R satisfying
|
R
|
<
3
L
n
−
5
{\displaystyle \vert R\vert <3L_{n}^{-5}}
.
Lucas identities
Many of the Fibonacci identities have parallels in Lucas numbers. For example, the Cassini identity becomes
L
n
2
−
L
n
−
1
L
n
+
1
=
(
−
1
)
n
5
{\displaystyle L_{n}^{2}-L_{n-1}L_{n+1}=(-1)^{n}5}
Also
∑
k
=
0
n
L
k
=
L
n
+
2
−
1
{\displaystyle \sum _{k=0}^{n}L_{k}=L_{n+2}-1}
∑
k
=
0
n
L
k
2
=
L
n
L
n
+
1
+
2
{\displaystyle \sum _{k=0}^{n}L_{k}^{2}=L_{n}L_{n+1}+2}
2
L
n
−
1
2
+
L
n
2
=
L
2
n
+
1
+
5
F
n
−
2
2
{\displaystyle 2L_{n-1}^{2}+L_{n}^{2}=L_{2n+1}+5F_{n-2}^{2}}
where
F
n
=
L
n
−
1
+
L
n
+
1
5
{\displaystyle \textstyle F_{n}={\frac {L_{n-1}+L_{n+1}}{5}}}
.
L
n
k
=
∑
j
=
0
⌊
k
2
⌋
(
−
1
)
n
j
(
k
j
)
L
(
k
−
2
j
)
n
′
{\displaystyle L_{n}^{k}=\sum _{j=0}^{\lfloor {\frac {k}{2}}\rfloor }(-1)^{nj}{\binom {k}{j}}L'_{(k-2j)n}}
where
L
n
′
=
L
n
{\displaystyle L'_{n}=L_{n}}
except for
L
0
′
=
1
{\displaystyle L'_{0}=1}
.
For example if n is odd,
L
n
3
=
L
3
n
′
−
3
L
n
′
{\displaystyle L_{n}^{3}=L'_{3n}-3L'_{n}}
and
L
n
4
=
L
4
n
′
−
4
L
2
n
′
+
6
L
0
′
{\displaystyle L_{n}^{4}=L'_{4n}-4L'_{2n}+6L'_{0}}
Checking,
L
3
=
4
,
4
3
=
64
=
76
−
3
(
4
)
{\displaystyle L_{3}=4,4^{3}=64=76-3(4)}
, and
256
=
322
−
4
(
18
)
+
6
{\displaystyle 256=322-4(18)+6}
Generating function
Let
Φ
(
x
)
=
2
+
x
+
3
x
2
+
4
x
3
+
⋯
=
∑
n
=
0
∞
L
n
x
n
{\displaystyle \Phi (x)=2+x+3x^{2}+4x^{3}+\cdots =\sum _{n=0}^{\infty }L_{n}x^{n}}
be the generating function of the Lucas numbers. By a direct computation,
Φ
(
x
)
=
L
0
+
L
1
x
+
∑
n
=
2
∞
L
n
x
n
=
2
+
x
+
∑
n
=
2
∞
(
L
n
−
1
+
L
n
−
2
)
x
n
=
2
+
x
+
∑
n
=
1
∞
L
n
x
n
+
1
+
∑
n
=
0
∞
L
n
x
n
+
2
=
2
+
x
+
x
(
Φ
(
x
)
−
2
)
+
x
2
Φ
(
x
)
{\displaystyle {\begin{aligned}\Phi (x)&=L_{0}+L_{1}x+\sum _{n=2}^{\infty }L_{n}x^{n}\\&=2+x+\sum _{n=2}^{\infty }(L_{n-1}+L_{n-2})x^{n}\\&=2+x+\sum _{n=1}^{\infty }L_{n}x^{n+1}+\sum _{n=0}^{\infty }L_{n}x^{n+2}\\&=2+x+x(\Phi (x)-2)+x^{2}\Phi (x)\end{aligned}}}
which can be rearranged as
Φ
(
x
)
=
2
−
x
1
−
x
−
x
2
{\displaystyle \Phi (x)={\frac {2-x}{1-x-x^{2}}}}
Φ
(
−
1
x
)
{\displaystyle \Phi \!\left(-{\frac {1}{x}}\right)}
gives the generating function for the negative indexed Lucas numbers,
∑
n
=
0
∞
(
−
1
)
n
L
n
x
−
n
=
∑
n
=
0
∞
L
−
n
x
−
n
{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}L_{n}x^{-n}=\sum _{n=0}^{\infty }L_{-n}x^{-n}}
, and
Φ
(
−
1
x
)
=
x
+
2
x
2
1
−
x
−
x
2
{\displaystyle \Phi \!\left(-{\frac {1}{x}}\right)={\frac {x+2x^{2}}{1-x-x^{2}}}}
Φ
(
x
)
{\displaystyle \Phi (x)}
satisfies the functional equation
Φ
(
x
)
−
Φ
(
−
1
x
)
=
2
{\displaystyle \Phi (x)-\Phi \!\left(-{\frac {1}{x}}\right)=2}
As the generating function for the Fibonacci numbers is given by
s
(
x
)
=
x
1
−
x
−
x
2
{\displaystyle s(x)={\frac {x}{1-x-x^{2}}}}
we have
s
(
x
)
+
Φ
(
x
)
=
2
1
−
x
−
x
2
{\displaystyle s(x)+\Phi (x)={\frac {2}{1-x-x^{2}}}}
which proves that
F
n
+
L
n
=
2
F
n
+
1
,
{\displaystyle F_{n}+L_{n}=2F_{n+1},}
and
5
s
(
x
)
+
Φ
(
x
)
=
2
x
Φ
(
−
1
x
)
=
2
1
1
−
x
−
x
2
+
4
x
1
−
x
−
x
2
{\displaystyle 5s(x)+\Phi (x)={\frac {2}{x}}\Phi (-{\frac {1}{x}})=2{\frac {1}{1-x-x^{2}}}+4{\frac {x}{1-x-x^{2}}}}
proves that
5
F
n
+
L
n
=
2
L
n
+
1
{\displaystyle 5F_{n}+L_{n}=2L_{n+1}}
The partial fraction decomposition is given by
Φ
(
x
)
=
1
1
−
ϕ
x
+
1
1
−
ψ
x
{\displaystyle \Phi (x)={\frac {1}{1-\phi x}}+{\frac {1}{1-\psi x}}}
where
ϕ
=
1
+
5
2
{\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}
is the golden ratio and
ψ
=
1
−
5
2
{\displaystyle \psi ={\frac {1-{\sqrt {5}}}{2}}}
is its conjugate.
This can be used to prove the generating function, as
∑
n
=
0
∞
L
n
x
n
=
∑
n
=
0
∞
(
ϕ
n
+
ψ
n
)
x
n
=
∑
n
=
0
∞
ϕ
n
x
n
+
∑
n
=
0
∞
ψ
n
x
n
=
1
1
−
ϕ
x
+
1
1
−
ψ
x
=
Φ
(
x
)
{\displaystyle \sum _{n=0}^{\infty }L_{n}x^{n}=\sum _{n=0}^{\infty }(\phi ^{n}+\psi ^{n})x^{n}=\sum _{n=0}^{\infty }\phi ^{n}x^{n}+\sum _{n=0}^{\infty }\psi ^{n}x^{n}={\frac {1}{1-\phi x}}+{\frac {1}{1-\psi x}}=\Phi (x)}
Congruence relations
If
F
n
≥
5
{\displaystyle F_{n}\geq 5}
is a Fibonacci number then no Lucas number is divisible by
F
n
{\displaystyle F_{n}}
.
The Lucas numbers satisfy Gauss congruence. This implies that
L
n
{\displaystyle L_{n}}
is congruent to 1 modulo
n
{\displaystyle n}
if
n
{\displaystyle n}
is prime. The composite values of
n
{\displaystyle n}
which satisfy this property are known as Fibonacci pseudoprimes.
L
n
−
L
n
−
4
{\displaystyle L_{n}-L_{n-4}}
is congruent to 0 modulo 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).
As of September 2015, the largest confirmed Lucas prime is L148091, which has 30950 decimal digits. As of August 2022, the largest known Lucas probable prime is L5466311, with 1,142,392 decimal digits.
If Ln is prime then n is 0, prime, or a power of 2. L2m is prime for m = 1, 2, 3, and 4 and no other known values of m.
Lucas polynomials
In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials
L
n
(
x
)
{\displaystyle L_{n}(x)}
are a polynomial sequence derived from the Lucas numbers.
Continued fractions for powers of the golden ratio
Close rational approximations for powers of the golden ratio can be obtained from their continued fractions.
For positive integers n, the continued fractions are:
φ
2
n
−
1
=
[
L
2
n
−
1
;
L
2
n
−
1
,
L
2
n
−
1
,
L
2
n
−
1
,
…
]
{\displaystyle \varphi ^{2n-1}=[L_{2n-1};L_{2n-1},L_{2n-1},L_{2n-1},\ldots ]}
φ
2
n
=
[
L
2
n
−
1
;
1
,
L
2
n
−
2
,
1
,
L
2
n
−
2
,
1
,
L
2
n
−
2
,
1
,
…
]
{\displaystyle \varphi ^{2n}=[L_{2n}-1;1,L_{2n}-2,1,L_{2n}-2,1,L_{2n}-2,1,\ldots ]}
.
For example:
φ
5
=
[
11
;
11
,
11
,
11
,
…
]
{\displaystyle \varphi ^{5}=[11;11,11,11,\ldots ]}
is the limit of
11
1
,
122
11
,
1353
122
,
15005
1353
,
…
{\displaystyle {\frac {11}{1}},{\frac {122}{11}},{\frac {1353}{122}},{\frac {15005}{1353}},\ldots }
with the error in each term being about 1% of the error in the previous term; and
φ
6
=
[
18
−
1
;
1
,
18
−
2
,
1
,
18
−
2
,
1
,
18
−
2
,
1
,
…
]
=
[
17
;
1
,
16
,
1
,
16
,
1
,
16
,
1
,
…
]
{\displaystyle \varphi ^{6}=[18-1;1,18-2,1,18-2,1,18-2,1,\ldots ]=[17;1,16,1,16,1,16,1,\ldots ]}
is the limit of
17
1
,
18
1
,
305
17
,
323
18
,
5473
305
,
5796
323
,
98209
5473
,
104005
5796
,
…
{\displaystyle {\frac {17}{1}},{\frac {18}{1}},{\frac {305}{17}},{\frac {323}{18}},{\frac {5473}{305}},{\frac {5796}{323}},{\frac {98209}{5473}},{\frac {104005}{5796}},\ldots }
with the error in each term being about 0.3% that of the second previous term.
Applications
Lucas numbers are the second most common pattern in sunflowers after Fibonacci numbers, when clockwise and counter-clockwise spirals are counted, according to an analysis of 657 sunflowers in 2016.
See also
Generalizations of Fibonacci numbers
References
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)
Kata Kunci Pencarian:
Lucas Numbers — Numberphile
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
Lucas Number -- from Wolfram MathWorld
lucas number
Daftar Isi
Lucas number - Wikipedia
The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers .
The First 200 Lucas numbers and their factors - University of Surrey
This page follows on from an Introduction to the Lucas Numbers. They are a variation on The Fibonacci Numbers. The Lucas numbers are defined very similarly to the Fibonacci numbers, but start with 2 and 1 (in this order) rather than the Fibonacci's 0 and 1:
Lucas Numbers - GeeksforGeeks
Oct 29, 2024 · Lucas numbers are similar to Fibonacci numbers. Lucas numbers are also defined as the sum of its two immediately previous terms. But here the first two terms are 2 and 1 whereas in Fibonacci numbers the first two terms are 0 and 1 respectively.
Lucas Number -- from Wolfram MathWorld
The Lucas numbers are the sequence of integers {L_n}_(n=1)^infty defined by the linear recurrence equation L_n=L_(n-1)+L_(n-2) (1) with L_1=1 and L_2=3. The nth Lucas number is implemented in the Wolfram Language as LucasL[n]. The values of L_n for n=1, 2, ... are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).
Lucas Numbers | Brilliant Math & Science Wiki
The Lucas numbers or Lucas series are an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–91), who studied both that sequence and the closely related Fibonacci numbers.
Lucas Sequence Calculator - Check Lucas Numbers & Calculate …
What is a Lucas Sequence? The Lucas Sequence is a type of recursive number series defined as: \( L_0 = 2 \) (0th term) \( L_1 = 1 \) (1st term) For \( n \geq 2 \), \( L_n = L_{n-1} + L_{n-2} \) (each term is the sum of the two preceding terms). Numbers in this sequence are called Lucas numbers. How to Check if a Number is a Lucas Number
Golden Ratio, Fibonacci Numbers and Lucas Numbers
Nov 7, 1999 · We can write a general formula to generate a Fibonnaci sequence using. If we take the ratio of two successive numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ..) and we divide each by the number before it, we will find the following series of numbers:
Introduction to the Fibonacci and Lucas numbers - Wolfram
Fibonacci and Lucas numbers have numerous applications throughout algebraic coding theory, linear sequential circuits, quasicrystals, phyllotaxies, biomathematics, and computer science.
Lucas Series Calculator - DQYDJ
A Lucas number is a number which appears in the Lucas sequence or alternatively the index of a number in the series. For example, the 6th Lucas number is 18, and 18 is also a Lucas number since it's in the series.
A000032 - OEIS
For n >= 3, the Lucas number L(n) is the dimension of a commutative Hecke algebra of affine type A_n with independent parameters. See Theorem 1.4, Corollary 1.5, and the table on page 524 in the link "Hecke algebras with independent parameters". - Jia Huang, Jan 20 2019