A335059 a(n) is the number of vertices in an n-gon formed by the straight line segments connecting vertex k to vertex 2k mod n.

3, 4, 6, 7, 11, 11, 14, 16, 26, 22, 36, 33, 40, 45, 61, 50, 76, 72, 81, 87, 111, 95, 131, 124, 137, 146, 176, 145, 201, 193, 208, 218, 256, 228, 286, 275, 294, 307, 351, 316, 386, 374, 395, 409, 461, 421, 501, 486, 511, 528, 586, 539, 631, 615, 642, 660, 726

Offset: 3

Author

Lars Blomberg, May 24 2020

Keywords

• nonn

Comments

See A335057 for illustrations.

Links

• Lars Blomberg, Table of n, a(n) for n = 3..270

Crossrefs

Cf. A335057 (regions), A335058 (edges), A335129 (distinct lines).

Programs

PARI

bc=[[5,0,19,24],[5,-16,75,24],[5,-18,64,24],[5,-18,88,24],[5,-34,24,24],[5,-34,192,24],[5,-34,216,24],[5,-34,0,24]];
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)=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.

Select the row in the table below for which d = n mod m. Then a(n) = (a*n^2+bn+c)/denom.

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

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

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

| 1, 5 | 6 | 5 | 0 | 19 | 24 |

| 3 | 6 | 5 | -16 | 75 | 24 |

| 2, 10 | 12 | 5 | -18 | 64 | 24 |

| 4, 8 | 12 | 5 | -18 | 88 | 24 |

| 0 | 60 | 5 | -34 | 24 | 24 |

| 6, 18, 42, 54 | 60 | 5 | -34 | 192 | 24 |

| 12, 24, 36, 48 | 60 | 5 | -34 | 216 | 24 |

| 30 | 60 | 5 | -34 | 0 | 24 |

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