This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A373325 #20 Jun 14 2024 16:29:55 %S A373325 1,2,10,66,498,4072,35144,315352,2914074,27553880,265387528,2595131328 %N A373325 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 A373325 Luc Rousseau, <a href="/A373325/a373325.jpg">Illustration for n=0..3</a> %H A373325 Luc Rousseau, <a href="/A373325/a373325.pl.txt">A Prolog program.</a> %e A373325 Curves without self-intersection are equivalent; one might for instance take the half-line y <= 0 as their representative; so a(0) = 1. %e A373325 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 A373325 When "outside" it might help visualization to imagine that a noncrossable oriented edge "at infinity" closes the cycle. %e A373325 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 A373325 See provided illustration for n=0..3 in section 'Links'. %o A373325 (SWI-Prolog) % see link. %Y A373325 Cf. A000682, A268563. %K A373325 nonn,more %O A373325 0,2 %A A373325 _Luc Rousseau_, Jun 01 2024