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A335057 a(n) is the number of regions inside an n-gon formed by the straight line segments connecting vertex k to vertex 2k mod n.

%I A335057 #10 May 24 2020 17:48:08
%S A335057 1,2,4,4,9,10,11,16,24,20,34,34,37,46,59,50,74,74,78,90,109,96,129,
%T A335057 128,134,150,174,152,199,198,205,224,254,232,284,282,291,314,349,322,
%U A335057 384,382,392,418,459,428,499,496,508,538,584,548,629,626,639,672,724
%N A335057 a(n) is the number of regions inside an n-gon formed by the straight line segments connecting vertex k to vertex 2k mod n.
%C A335057 The envelope of the lines form a cardioid.
%H A335057 Lars Blomberg, <a href="/A335057/b335057.txt">Table of n, a(n) for n = 3..270</a>
%H A335057 Lars Blomberg, <a href="/A335057/a335057.png">Illustration for n = 12</a>
%H A335057 Lars Blomberg, <a href="/A335057/a335057_1.png">Illustration for n = 18</a>
%H A335057 Lars Blomberg, <a href="/A335057/a335057_2.png">Illustration for n = 32</a>
%H A335057 Lars Blomberg, <a href="/A335057/a335057_3.png">Illustration for n = 107</a>
%F A335057 Empirically for n <= 270.
%F A335057 For n > 3 select the row in the table below for which d = n mod m. Then a(n) = (a*n^2+b*n+c)/denom.
%F A335057 +=============================================+
%F A335057 |              d |  m | a |   b |   c | denom |
%F A335057 +---------------------------------------------+
%F A335057 |           1, 5 |  6 | 5 |   0 | -29 |    24 |
%F A335057 |              3 |  6 | 5 | -16 |   3 |    24 |
%F A335057 |          2, 10 | 12 | 5 | -12 |   4 |    24 |
%F A335057 |           4, 8 | 12 | 5 | -12 |  16 |    24 |
%F A335057 |              0 | 60 | 5 | -28 |   0 |    24 |
%F A335057 |  6, 18, 42, 54 | 60 | 5 | -28 |  84 |    24 |
%F A335057 | 12, 24, 36, 48 | 60 | 5 | -28 |  96 |    24 |
%F A335057 |             30 | 60 | 5 | -28 | -12 |    24 |
%F A335057 +=============================================+
%o A335057 (PARI) bc=[[5,0,-29,24],[5,-16,3,24],[5,-12,4,24],[5,-12,16,24],[5,-28,0,24],[5,-28,84,24],[5,-28,96,24],[5,-28,-12,24]];
%o A335057 m=[[1,6,1],[5,6,1],[3,6,2],[2,12,3],[10,12,3],[4,12,4],[8,12,4],[0,60,5],[6,60,6],[18,60,6],[42,60,6],[54,60,6],[12,60,7],[24,60,7],[36,60,7],[48,60,7],[30,60,8]];
%o A335057 ix(n)=for(i=1,length(m),x=m[i];if(n%x[2]==x[1], return(x[3])));-1
%o A335057 a(n)=if(n==3,return(1));x=bc[ix(n)];(x[1]*n^2+x[2]*n+x[3])/x[4]
%o A335057 vector(200,x,a(x+2))
%Y A335057 Cf. A335058 (edges), A335059 (vertices), A335129 (distinct lines).
%K A335057 nonn
%O A335057 3,2
%A A335057 _Lars Blomberg_, May 23 2020