A373325 - OEIS

%I #20 Jun 14 2024 16:29:55

%S 1,2,10,66,498,4072,35144,315352,2914074,27553880,265387528,2595131328

%N Number of semi-infinite curves of the plane with n simple, transverse self-intersections and no other self-intersections, up to an orientation-preserving homeomorphism.

%H Luc Rousseau, <a href="/A373325/a373325.jpg">Illustration for n=0..3</a>

%H Luc Rousseau, <a href="/A373325/a373325.pl.txt">A Prolog program.</a>

%e Curves without self-intersection are equivalent; one might for instance take the half-line y <= 0 as their representative; so a(0) = 1.

%e 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.

%e When "outside" it might help visualization to imagine that a noncrossable oriented edge "at infinity" closes the cycle.

%e 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.

%e See provided illustration for n=0..3 in section 'Links'.

%o (SWI-Prolog) % see link.

%Y Cf. A000682, A268563.

%K nonn,more

%O 0,2

%A _Luc Rousseau_, Jun 01 2024