A335058 a(n) is the number of edges in an n-gon formed by the straight line segments connecting vertex k to vertex 2k mod n.

3, 5, 9, 10, 19, 20, 24, 31, 49, 41, 69, 66, 76, 90, 119, 99, 149, 145, 158, 176, 219, 190, 259, 251, 270, 295, 349, 296, 399, 390, 412, 441, 509, 459, 569, 556, 584, 620, 699, 637, 769, 755, 786, 826, 919, 848, 999, 981, 1018, 1065, 1169, 1086, 1259, 1240

Offset

3,1

Comments

See A335057 for illustrations.

Links

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

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 | -17 | 12 |

| 3 | 6 | 5 | -16 | 27 | 12 |

| 2, 10 | 12 | 5 | -15 | 22 | 12 |

| 4, 8 | 12 | 5 | -15 | 40 | 12 |

| 0 | 60 | 5 | -31 | 0 | 12 |

| 6, 18, 42, 54 | 60 | 5 | -31 | 126 | 12 |

| 12, 24, 36, 48 | 60 | 5 | -31 | 144 | 12 |

| 30 | 60 | 5 | -31 | -18 | 12 |

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

Prog

(PARI) bc=[[5, 0, -17, 12], [5, -16, 27, 12], [5, -15, 22, 12], [5, -15, 40, 12], [5, -31, 0, 12], [5, -31, 126, 12], [5, -31, 144, 12], [5, -31, -18, 12]];
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(3)); x=bc[ix(n)]; (x[1]*n^2+x[2]*n+x[3])/x[4]
vector(200, x, a(x+2))

Crossrefs

Cf. A335057 (regions), A335059 (vertices), A335129 (distinct lines).

Sequence in context: A063038 A236309 A304588 * A066769 A323765 A270780

Adjacent sequences: A335055 A335056 A335057 * A335059 A335060 A335061

Keyword

nonn

Author

Lars Blomberg, May 24 2020

Status

approved