## Lucas Numbers

The Lucas numbers
(A000204)
are illustrated by the following diagram:

(Figure drawn by Henry Bottomley, July 27 2000.)

If turned sideways, so that
the node labeled 1 is at the bottom, this may be regarded as the
**Lucas Tree**,
which grows according to the rules that

- initially there is one black node which
produces one blue node and two red nodes

- every red node turns blue after a year

- every blue node produces one blue node and one red node after a year

At the nth year there are L_{n} nodes.

Here is a different representation of the same tree.

This grows according to the rules that
every mature branch sprouts a new branch
at the end of each year, and new branches take
a year to reach maturity, except that in the first year
two new branches are formed instead of one.

Mature branches are indicated by heavy lines.

At the end of the nth year there are L_{n} branches.