The toothpick sequence counts the total number 
of line segments after
stages of the following growth process. Stage 0 is empty, and stage 1 consists of
one vertical unit line segment. An endpoint
is exposed when it is neither an endpoint nor a midpoint
of any other line segment. At each subsequent stage,
a new unit line segment is centered at every exposed
endpoint, perpendicular
to the existing line segment there. All additions
in a stage are made simultaneously (Applegate et al. 2010).
Starting with 
,
the totals are 0, 1, 3, 7, 11, 15, 23, 35, 43, 47, ... (OEIS A139250).
The numbers added at each stage are 
for 
, with 
. These begin 0, 1, 2, 4, 4, 4, 8, 12, 8, 4, ... (OEIS
A139251). They satisfy
 |
for nonnegative integers 
and
 |
for integers 
with 
(Applegate et al. 2010). The cumulative
totals are recovered as 
. This local growth rule provides a geometric
example related to cellular automata (Applegate
et al. 2010, Numberphile 2018).
