The Lucas numbers are the sequence of integers defined by the linear recurrence equation
with and . The th Lucas number is implemented in the Wolfram Language as LucasL[n].
The values of for , 2, ... are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).
The Lucas numbers are also a Lucas sequence and are the companions to the Fibonacci numbers and satisfy the same recurrence.
The number of ways of picking a set (including the empty set) from the numbers 1, 2, ..., without picking two consecutive numbers (where 1 and are now consecutive) is (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 is prime, . The converse does not necessarily hold true, however, and composite numbers such that are known as Lucas pseudoprimes.
For , 2, ..., the numbers of decimal digits in 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 (OEIS A097348), where is the golden ratio. This follows from the fact that for any power function , the number of decimal digits for is given by .
The lengths of the cycles for Lucas numbers (mod ) for , 2, ... are 12, 60, 300, 3000, 30000, 300000, 300000, ... (OEIS A114307).
The analog of Binet's Fibonacci numberformula for Lucas numbers is
Another formula is
for , where is the golden ratio and denotes the nearest integer function.
Another recurrence relation for is given by,
for , where is the floor function.
Additional identities satisfied by Lucas numbers include
The Lucas numbers obey the negation formula
the addition formula
where is a Fibonacci number, the subtraction formula
the fundamental identity
and power expansion
The Lucas numbers satisfy the power recurrence
where is a Fibonomial coefficient, the reciprocal sum
the partial fraction decomposition
and the summation formula
Let be a prime and be a positive integer. Then ends in a 3 (Honsberger 1985, p. 113). Analogs of the Cesàro identities for Fibonacci numbers are
where is a binomial coefficient.
( divides ) iff divides into an even number of times. iff divides into an odd number of times. always ends in 2 (Honsberger 1985, p. 137).
(Honsberger 1985, pp. 113-114).