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.

1, 2, 4, 4, 9, 10, 11, 16, 24, 20, 34, 34, 37, 46, 59, 50, 74, 74, 78, 90, 109, 96, 129, 128, 134, 150, 174, 152, 199, 198, 205, 224, 254, 232, 284, 282, 291, 314, 349, 322, 384, 382, 392, 418, 459, 428, 499, 496, 508, 538, 584, 548, 629, 626, 639, 672, 724

Offset: 3

Author

Lars Blomberg, May 23 2020

Keywords

• nonn

Comments

The envelope of the lines form a cardioid.

Links

• Lars Blomberg, Table of n, a(n) for n = 3..270
• Lars Blomberg, Illustration for n = 12
• Lars Blomberg, Illustration for n = 18
• Lars Blomberg, Illustration for n = 32
• Lars Blomberg, Illustration for n = 107

Crossrefs

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

Programs

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]];
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]];
ix(n)=for(i=1,length(m),x=m[i];if(n%x[2]==x[1], return(x[3])));-1
a(n)=if(n==3,return(1));x=bc[ix(n)];(x[1]*n^2+x[2]*n+x[3])/x[4]
vector(200,x,a(x+2))

Formula

Empirically for n <= 270.

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.

+=============================================+

| d | m | a | b | c | denom |

+---------------------------------------------+

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

| 3 | 6 | 5 | -16 | 3 | 24 |

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

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

| 0 | 60 | 5 | -28 | 0 | 24 |

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

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

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

+=============================================+