Similar to A139250, except that when we add toothpicks to horizontal toothpicks, we only add them at the left-hand end.
Sequence gives total number of toothpicks in the n-th generation. First differences are in A060632.
This is equivalent to the Sierpinski triangle A047999. Each inverted T formed by two toothpicks is equivalent to a triangle in the Sierpinski sieve. See Gould's sequence A001316. [From Omar E. Pol, May 23 2009]