with

*
and
*
. The
*
th Lucas number is imposed in the rebab.net Language together LucasL.

You are watching: 1 3 4 7 11 18 29

The worths of

*
because that
*
, 2, ... Room 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ... (OEIS A000204).

The Lucas numbers are additionally a Lucas succession

*
and also are the companions come the Fibonacci number
*
and also satisfy the very same recurrence.

The number of ways of picking a collection (including the north set) indigenous the numbers 1, 2, ...,

*
without picking 2 consecutive numbers (where 1 and also
*
are now consecutive) is
*
(Honsberger 1985, p.122).

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

Rather amazingly, if

*
is prime,
*
. The converse does no necessarily hold true, however, and composite number
*
such the
*
are known as Lucas pseudoprimes.

For

*
, 2, ..., the number of decimal digits in
*
*
(OEIS A097348), where
*
is the golden ratio. This adheres to from the fact that for any type of power function
*
, the number of decimal digits because that
*
is provided by
*
.

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The lengths the the cycles because that Lucas numbers (mod

*
) for
*
, 2, ... Are 12, 60, 300, 3000, 30000, 300000, 300000, ... (OEIS A114307).