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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. 9

%I

%S 1,2,4,4,9,10,11,16,24,20,34,34,37,46,59,50,74,74,78,90,109,96,129,

%T 128,134,150,174,152,199,198,205,224,254,232,284,282,291,314,349,322,

%U 384,382,392,418,459,428,499,496,508,538,584,548,629,626,639,672,724

%N 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 The envelope of the lines form a cardioid.

%H Lars Blomberg, <a href="/A335057/b335057.txt">Table of n, a(n) for n = 3..270</a>

%H Lars Blomberg, <a href="/A335057/a335057.png">Illustration for n = 12</a>

%H Lars Blomberg, <a href="/A335057/a335057_1.png">Illustration for n = 18</a>

%H Lars Blomberg, <a href="/A335057/a335057_2.png">Illustration for n = 32</a>

%H Lars Blomberg, <a href="/A335057/a335057_3.png">Illustration for n = 107</a>

%F Empirically for n <= 270.

%F 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 +=============================================+

%F | d | m | a | b | c | denom |

%F +---------------------------------------------+

%F | 1, 5 | 6 | 5 | 0 | -29 | 24 |

%F | 3 | 6 | 5 | -16 | 3 | 24 |

%F | 2, 10 | 12 | 5 | -12 | 4 | 24 |

%F | 4, 8 | 12 | 5 | -12 | 16 | 24 |

%F | 0 | 60 | 5 | -28 | 0 | 24 |

%F | 6, 18, 42, 54 | 60 | 5 | -28 | 84 | 24 |

%F | 12, 24, 36, 48 | 60 | 5 | -28 | 96 | 24 |

%F | 30 | 60 | 5 | -28 | -12 | 24 |

%F +=============================================+

%o (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 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 ix(n)=for(i=1,length(m),x=m[i];if(n%x[2]==x[1], return(x[3])));-1

%o a(n)=if(n==3,return(1));x=bc[ix(n)];(x[1]*n^2+x[2]*n+x[3])/x[4]

%o vector(200,x,a(x+2))

%Y Cf. A335058 (edges), A335059 (vertices), A335129 (distinct lines).

%K nonn

%O 3,2

%A _Lars Blomberg_, May 23 2020

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Last modified August 4 03:18 EDT 2021. Contains 346442 sequences. (Running on oeis4.)