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

3,1

Comments

See A335057 for illustrations.

Links

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

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 |

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

Prog

(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))

Crossrefs

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

Sequence in context: A064404 A294488 A364341 * A047514 A345716 A011975

Adjacent sequences: A335056 A335057 A335058 * A335060 A335061 A335062

Keyword

nonn

Author

Lars Blomberg, May 24 2020

Status

approved