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Lucas Number

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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).

The Lucas numbers are also a Lucas sequence V_n(1,-1) and are the companions to the Fibonacci numbers F_n and satisfy the same recurrence.

The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., n without picking two consecutive numbers (where 1 and n are now consecutive) is L_n (Honsberger 1985, p. 122).

The only square numbers in the Lucas sequence are 1 and 4 (Alfred 1964, Cohn 1964). The only triangular Lucas numbers are 1, 3, and 5778 (Ming 1991). The only cubic Lucas number is 1.

Rather amazingly, if n is prime, L_n=1 (mod n). The converse does not necessarily hold true, however, and composite numbers n such that L_n=1 (mod n) are known as Lucas pseudoprimes.

For n=1, 2, ..., the numbers of decimal digits in L_(10^n) are 1, 3, 21, 209, 2090, 20899, 208988, 2089877, ... (OEIS A114469). As can be seen, the initial strings of digits settle down to produce the number 208987640249978733769..., which corresponds to the decimal digits of lnphi=0.2089876... (OEIS A097348), where phi is the golden ratio. This follows from the fact that for any power function f_n=c^n, the number of decimal digits for f_(10^n) is given by 10^nlog_(10)c.

The lengths of the cycles for Lucas numbers (mod 10^n) for n=1, 2, ... are 12, 60, 300, 3000, 30000, 300000, 300000, ... (OEIS A114307).

The analog of Binet's formula for Lucas numbers is

L_n=((1+sqrt(5))/2)^n+((1-sqrt(5))/2)^n.
(2)

Another formula is

L_n=[phi^n]
(3)

for n>=2, where phi is the golden ratio and [x] denotes the nearest integer function.

Another recurrence relation for L_n is given by,

L_(n+1)=|_(L_n(1+sqrt(5))+1)/2_|
(4)

for n>=4, where |_x_| is the floor function.

Additional identities satisfied by Lucas numbers include

L_n^2-L_(n-1)L_(n+1)=5(-1)^n,
(5)

and

sum_(k=1)^nL_k^2=L_nL_(n+1)-2.
(6)

The Lucas numbers obey the negation formula

L_(-n)=(-1)^nL_n,
(7)

the addition formula

L_(m+n)=1/2(5F_mF_n+L_mL_n),
(8)

where F_n is a Fibonacci number, the subtraction formula

L_(m-n)=1/2(-1)^n(L_mL_n-5F_mF_n),
(9)

the fundamental identity

L_n^2-5F_n^2=4(-1)^n,
(10)

conjugation relation

L_n=F_(n-1)+F_(n+1),
(11)

successor relation

L_(n+1)=1/2(5F_n+L_n),
(12)

double-angle formula

L_(2n)=1/2(5F_n^2+L_n^2),
(13)

multiple-angle recurrence

L_(kn)=L_kL_(k(n-1))-(-1)^kL_(k(n-2)),
(14)

multiple-angle formulas

L_(kn)=1/(2^(k-1))sum_(i=0)^(|_k/2_|)(k; 2i)5^iF_n^(2i)L_n^(k-2i)
(15)
=sum_(i=0)^(|_k/2_|)k/(k-i)(k-i; i)(-1)^(i(n+1))L_n^(k-2i)
(16)
={sum_(i=0)^(k/2)k/(k-i)(k-i; i)(-1)^(in)5^(k/2-i)F_n^(k-2i) for k even; L_nsum_(i=0)^(|_k/2_|)(k-1-i; i)(-1)^(in)5^(|_k/2_|-i)F_n^(k-1-2i) for k odd
(17)
=sum_(i=0)^(k)(k; i)L_iF_n^iF_(n-1)^(k-i),
(18)

product expansions

F_mL_n=F_(m+n)+(-1)^nF_(m-n)
(19)

and

F_mF_n=1/5[L_(m+n)-(-1)^nL_(m-n)],
(20)

square expansion,

L_n^2=L_(2n)+2(-1)^n,
(21)

and power expansion

L_n^k=1/2sum_(i=0)^k(k; i)(-1)^(in)L_((k-2i)n).
(22)

The Lucas numbers satisfy the power recurrence

sum_(j=0)^(t+1)(-1)^(j(j+1)/2)[t+1; j]_FL_(n-j)^t=0,
(23)

where [a; b]_F is a Fibonomial coefficient, the reciprocal sum

sum_(k=1)^n((-1)^k)/(L_kL_(k+a))=(F_n)/(F_a)sum_(k=1)^a((-1)^k)/(L_kL_(k+n)),
(24)

the convolution

sum_(k=0)^nL_kL_(n-k)=(n+2)L_n+F_n,
(25)

the partial fraction decomposition

-5/(L_(n+a)L_(n+b)L_(n+c))=A/(L_(n+a))+B/(L_(n+b))+C/(L_(n+c)),
(26)

where

A=((-1)^(n-a))/(F_(b-a)F_(c-a))
(27)
B=((-1)^(n-b))/(F_(c-b)F_(a-b))
(28)
C=((-1)^(n-c))/(F_(a-c)F_(b-c)),
(29)

and the summation formula

sum_(k=0)^nx^kL_(ak+b)=(g(n+1)-g(0))/(1-L_ax+(-1)^ax^2),
(30)

where

g(n)=(-1)^aL_(a(n-1)+b)x^(n+1)-L_(an+b)x^n.
(31)

Let p be a prime >3 and k be a positive integer. Then L_(2p^k) ends in a 3 (Honsberger 1985, p. 113). Analogs of the Cesàro identities for Fibonacci numbers are

sum_(k=0)^n(n; k)L_k=L_(2n)
(32)
sum_(k=0)^n(n; k)2^kL_k=L_(3n),
(33)

where (n; k) is a binomial coefficient.

L_n|F_m (L_n divides F_m) iff n divides into m an even number of times. L_n|L_m iff n divides into m an odd number of times. 2^nL_n always ends in 2 (Honsberger 1985, p. 137).

Defining

D_n=|3 i 0 0 ... 0 0; i 1 i 0 ... 0 0; 0 i 1 i ... 0 0; 0 0 i 1 ... 0 0; | | | | ... | |; 0 0 0 0 ... 1 i; 0 0 0 0 ... i 1|=L_(n+1)
(34)

gives

D_n=D_(n-1)+D_(n-2)
(35)

(Honsberger 1985, pp. 113-114).

See also

Fibonacci Number, Integer Sequence Primes, Lucas n-Step Number, Lucas Polynomial, Lucas Prime, Lucas Pseudoprime, Lucas Sequence, Reciprocal Fibonacci Constant

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References

Alfred, Brother U. "On Square Lucas Numbers." Fib. Quart. 2, 11-12, 1964.Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, pp. 94-101, 1987.Brillhart, J.; Montgomery, P. L.; and Silverman, R. D. "Tables of Fibonacci and Lucas Factorizations." Math. Comput. 50, 251-260 and S1-S15, 1988.Broadhurst, D. and Irvine, S. "Lucas Record." Post to primeform user forum. Jun. 19, 2006. http://groups.yahoo.com/group/primeform/message/7534.Brown, J. L. Jr. "Unique Representation of Integers as Sums of Distinct Lucas Numbers." Fib. Quart. 7, 243-252, 1969.Cohn, J. H. E. "Square Fibonacci Numbers, etc." Fib. Quart. 2, 109-113, 1964.Dubner, H. and Keller, W. "New Fibonacci and Lucas Primes." Math. Comput. 68, 417-427 and S1-S12, 1999.Guy, R. K. "Fibonacci Numbers of Various Shapes." §D26 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 194-195, 1994.Hilton, P.; Holton, D.; and Pedersen, J. "Fibonacci and Lucas Numbers." Ch. 3 in Mathematical Reflections in a Room with Many Mirrors. New York: Springer-Verlag, pp. 61-85, 1997.Hilton, P. and Pedersen, J. "Fibonacci and Lucas Numbers in Teaching and Research." J. Math. Informatique 3, 36-57, 1991-1992.Hoggatt, V. E. Jr. The Fibonacci and Lucas Numbers. Boston, MA: Houghton Mifflin, 1969.Honsberger, R. "A Second Look at the Fibonacci and Lucas Numbers." Ch. 8 in Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985.Koshy, T. Fibonacci and Lucas Numbers with Applications. New York: Wiley-Interscience, 2001.Update a linkLeyland, P. ftp://sable.ox.ac.uk/pub/math/factors/lucas.ZLifchitz, H. and Lifchitz, R. "PRP Top Records." http://www.primenumbers.net/prptop/searchform.php?form=L(n).Ming, L. "On Triangular Lucas Numbers." Applications of Fibonacci Numbers, Vol. 4 (Ed. G. E. Bergum, A. N. Philippou, and A. F. Horadam). Dordrecht, Netherlands: Kluwer, pp. 231-240, 1991.Sloane, N. J. A. Sequences A000204/M2341, A001606/M0961, A005479/M2627, A068070, A097348, A114469, and A114307 in "The On-Line Encyclopedia of Integer Sequences."

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Lucas Number

Cite this as:

Weisstein, Eric W. "Lucas Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LucasNumber.html

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