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This sequence is an E-toothpick sequence (cf. A161328) but starting with two back-to-back E-toothpicks.

On the infinite triangular grid, we start at round 0 with no E-toothpicks.

At round 1 we place two back-to-back E-toothpicks, forming a star with six endpoints.

At round 2 we add six more E-toothpicks.

At round 3 we add six more E-toothpicks.

And so on ... (see the illustrations).

The rule for adding new E-toothpicks is as follows. Each E has three ends, which initially are free. If the ends of two E's meet, those ends are no longer free. To go from round n to round n+1, we add an E-toothpick at each free end (extending that end in the direction it is pointing), subject to the condition that no end of any new E can touch any end of an existing E from round n or earlier. (Two new E's are allowed to touch.)

The sequence gives the number of E-toothpicks in the structure after n rounds. A161331 (the first differences) gives the number added at the n-th round.

See the entry A139250 for more information about the toothpick process and the toothpick propagation.

Note that, on the infinite triangular grid, a E-toothpick can be represented as a polyedge with three components. In this case, at n-th round, the structure is a polyedge with 3*a(n) components.

An E-toothpick is formed by three toothpicks, as an trident. The E-toothpick has a midpoint and three exposed endpoints such that the distance between the endpoint of the central toothpick and the endpoints of the other toothpicks is equal to 1.

On the infinite triangular grid, we start at round 0 with no E-toothpicks.

At round 1 we place an E-toothpick anywhere in the plane.

At round 2 we add three more E-toothpicks.

At round 3 we add five more E-toothpicks.

And so on... (see illustrations).

The rule for adding new E-toothpicks is as follows. Each E has three ends, which initially are free. If the ends of two E's meet, those ends are no longer free. To go from round n to round n+1, we add an E-toothpick at each free end (extending that end in the direction it is pointing), subject to the condition that no end of any new E can touch any end of an existing E from round n or earlier. (Two new E's are allowed to touch.)

The sequence gives the number of E-toothpicks in the structure after n rounds. A161329 (the first differences) gives the number added at the n-th round.

Note that, on the infinite triangular grid, a E-toothpick can be represented as a polyedge with three components. In this case, at n-th round, the structure is a polyedge with 3*a(n) components. See the entry A139250 for more information about the growth of the toothpicks.

See also the snowflake sequence A161330.

It appears that if n >> 1 the structure looks like an equilateral triangle, which is essentially one of the six wedges of the E-toothpick (or snowflake) structure of A161330. The sequence gives the number of E-toothpicks in the structure after n stages. A220498 (the first differences) gives the number added at the n-th round. For more information and some illustrations see A161330. For the E-toothpick right triangle see A211964.