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The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large. However, when its terms become very small, the arc's radius decreases rapidly from 3 to 1 then increases from 1 to 2.

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 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.^[1] 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.^[2]

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. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties.

The Lucas numbers may thus be defined as follows:

Ln:={2if n=0;1if n=1;Ln−1+Ln−2if n>1.displaystyle L_n:=begincases2&textif n=0;\1&textif n=1;\L_n-1+L_n-2&textif n>1.\endcases

(where n belongs to the natural numbers)

The sequence of Lucas numbers is:

2,1,3,4,7,11,18,29,47,76,123,…displaystyle 2,;1,;3,;4,;7,;11,;18,;29,;47,;76,;123,;ldots ;(sequence A000032 in the OEIS).

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, 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 Lndisplaystyle L_n for −5≤n≤5displaystyle -5leq nleq 5 are shown).

The formula for terms with negative indices in this sequence is

L−n=(−1)nLn.displaystyle L_-n=(-1)^nL_n.!

Relationship to Fibonacci numbers

The first identity expressed visually

The Lucas numbers are related to the Fibonacci numbers by the identities

• Ln=Fn−1+Fn+1=Fn+2Fn−1=Fn+2−Fn−2displaystyle ,L_n=F_n-1+F_n+1=F_n+2F_n-1=F_n+2-F_n-2


• Lm+n=Lm+1Fn+LmFn−1displaystyle ,L_m+n=L_m+1F_n+L_mF_n-1


• 
  Ln2=5Fn2+4(−1)ndisplaystyle ,L_n^2=5F_n^2+4(-1)^n, and thus as ndisplaystyle n, approaches +∞, the ratio LnFndisplaystyle frac L_nF_n approaches 5.displaystyle sqrt 5.
  


• F2n=LnFndisplaystyle ,F_2n=L_nF_n


• Fn+k+(−1)kFn−k=LkFndisplaystyle ,F_n+k+(-1)^kF_n-k=L_kF_n


• Fn=Ln−1+Ln+15=Ln−3+Ln+310displaystyle ,F_n=L_n-1+L_n+1 over 5=L_n-3+L_n+3 over 10

Their closed formula is given as:

Ln=φn+(1−φ)n=φn+(−φ)−n=(1+52)n+(1−52)n,displaystyle L_n=varphi ^n+(1-varphi )^n=varphi ^n+(-varphi )^-n=left(1+sqrt 5 over 2right)^n+left(1-sqrt 5 over 2right)^n,,

where φdisplaystyle varphi is the golden ratio. Alternatively, as for n>1displaystyle n>1 the magnitude of the term (−φ)−ndisplaystyle (-varphi )^-n is less than 1/2, Lndisplaystyle L_n is the closest integer to φndisplaystyle varphi ^n or, equivalently, the integer part of φn+1/2displaystyle varphi ^n+1/2, also written as ⌊φn+1/2⌋displaystyle lfloor varphi ^n+1/2rfloor .

Combining the above with Binet's formula,

Fn=φn−(1−φ)n5,displaystyle F_n=frac varphi ^n-(1-varphi )^nsqrt 5,,

a formula for φndisplaystyle varphi ^n is obtained:

φn=Ln+Fn52.displaystyle varphi ^n=L_n+F_nsqrt 5 over 2,.

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. These are the Bruckman-Lucas pseudoprimes.

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).

For these ns are

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 L_n is prime then n is either 0, prime, or a power of 2.^[3]L_2^m is prime for m = 1, 2, 3, and 4 and no other known values of m.

Generating series

Let

Φ(x)=2+x+3x2+4x3+⋯=∑n=0∞Lnxndisplaystyle Phi (x)=2+x+3x^2+4x^3+cdots =sum _n=0^infty L_nx^n

be the generating series of the Lucas numbers. By a direct computation,

Φ(x)=L0+L1x+∑n=2∞Lnxn=2+x+∑n=2∞(Ln−1+Ln−2)xn=2+x+∑n=1∞Lnxn+1+∑n=0∞Lnxn+2=2+x+x(Φ(x)−2)+x2Φ(x)displaystyle beginalignedPhi (x)&=L_0+L_1x+sum _n=2^infty L_nx^n\&=2+x+sum _n=2^infty (L_n-1+L_n-2)x^n\&=2+x+sum _n=1^infty L_nx^n+1+sum _n=0^infty L_nx^n+2\&=2+x+x(Phi (x)-2)+x^2Phi (x)endaligned

which can be rearranged as

Φ(x)=2−x1−x−x2.displaystyle Phi (x)=frac 2-x1-x-x^2.

The partial fraction decomposition is given by

Φ(x)=11−φx+11−ϕxdisplaystyle Phi (x)=frac 11-varphi x+frac 11-phi x

where φ=1+52displaystyle varphi =frac 1+sqrt 52 is the golden ratio and ϕ=1−52displaystyle phi =frac 1-sqrt 52 is its conjugate.

Lucas polynomials

In the same way as Fibonacci polynomials are derived from the Fibonacci numbers, the Lucas polynomials L_n(x) are a polynomial sequence derived from the Lucas numbers.

See also

• Generalizations of Fibonacci numbers

References

External links

• 
  Hazewinkel, Michiel, ed. (2001) [1994], "Lucas polynomials", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  


• 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)