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Curves without self-intersection are equivalent; one might for instance take the half-line y <= 0 as their representative; so a(0) = 1.
To get a curve with n+1 self-intersections, one can start from a curve with n self-intersections; identify the cycle of oriented edges that directly surrounds the finite extremity of the curve; choose an edge from that cycle and extend the curve so that it crosses that edge.
When "outside" it might help visualization to imagine that a noncrossable oriented edge "at infinity" closes the cycle.
Thus, for a transition between 0 and 1 self-intersection, the choice is between making a loop that turns left and making a loop that turns right; so a(1) = 2.
See provided illustration for n=0..3 in section 'Links'.
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