We start at stage 0 with no toothpicks.
At stage 1 we place a single toothpick touching a vertex of the infinite hexagon, in direction to the center of the hexagon, but on the outside corner, so a(1)=1.
At stage 2 we place 2 toothpicks touching the exposed endpoint of the initial toothpick, so a(2)=1+2=3.
At stage 3 we place 4 toothpicks, so a(3)=3+4=7.
At stage 4 we place 8 toothpicks, so a(4)=7+8=15.
At stage 5 we place 12 toothpicks, so a(5)=15+12=27.
After 5 stages the toothpick structure has 5 hexagons and 6 exposed endpoints.
