Lucas Number

The numbers produced by the $V$ recurrence in the Lucas Sequence with $(P,Q)=(1,-1)$ are called Lucas numbers. They are the companions to the Fibonacci Numbers $F_n$ and satisfy the same recurrence

$$L_n=L_{n-1}+L_{n-2},$$ (1)

where $L_1=1$, $L_2=3$. The first few are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (Sloane's A000204).

In terms of the Fibonacci Numbers,

$$L_n=F_{n-1}+F_{n+1}.$$ (2)

The analog of Binet's Formula for Lucas numbers is

$$L_n=\left({1+\sqrt{5}\over 2}\right)^n +\left({1-\sqrt{5}\over 2}\right)^n.$$ (3)

Another formula is

$$L_n=\left[{\phi^n}\right],$$ (4)

where $\phi$ is the Golden Ratio and $[x]$ denotes the Nint function. Given $L_{n}$,

$$L_{n+1}=\left\lfloor{{L_n}(1+\sqrt{5})+1\over 2}\right\rfloor ,$$ (5)

where $\left\lfloor{x}\right\rfloor$ is the Floor Function,

$${L_n}^2-L_{n-1}L_{n+1}=5(-1)^n,$$ (6)

and

$$\sum_{k=0}^n {L_k}^2=L_nL_{n+1}-2.$$ (7)

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

$$\sum_{k=0}^n {n\choose k} L_k = L_{2n}$$ (8)

$$\sum_{k=0}^n {n\choose k} 2^k L_k = L_{3n},$$ (9)

where ${n\choose k}$ is a Binomial Coefficient.

$L_n\vert F_m$ ($L_{n}$ Divides $F_m$) Iff $n$ Divides into $m$ an Even number of times. $L_n\vert L_m$ Iff $n$ divides into $m$ an Odd number of times. $2^nL_n$ always ends in 2 (Honsberger 1985, p. 137).

Defining

$$D_n\equiv \left\vert\matrix{ 3 & i & 0 & 0 & \cdots & 0 & 0... ... i\cr 0 & 0 & 0 & 0 & \cdots & i & 1\cr}\right\vert = L_{n+1}$$ (10)

gives

$$D_n=D_{n-1}+D_{n-2}$$ (11)

(Honsberger 1985, pp. 113-114).

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, as proved by John H. E. Cohn (Alfred 1964). The only Triangular Lucas numbers are 1, 3, and 5778 (Ming 1991). The only Lucas Cubic Number is 1. The first few Lucas Primes $L_{n}$ occur for $n=2$, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, ... (Dubner and Keller 1999, Sloane's A001606).

See also Fibonacci Number

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 Solverman, R. D. ``Tables of Fibonacci and Lucas Factorizations.'' Math. Comput. 50, 251-260 and S1-S15, 1988.

Brown, J. L. Jr. ``Unique Representation of Integers as Sums of Distinct Lucas Numbers.'' Fib. Quart. 7, 243-252, 1969.

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.

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.

Leyland, P. ftp://sable.ox.ac.uk/pub/math/factors/lucas.Z.

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 and A001606/M0961 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.