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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
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

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

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 |

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

PROG

(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))

CROSSREFS

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

Sequence in context: A233655 A307325 A272196 * A039887 A216162 A114215

Adjacent sequences:  A335054 A335055 A335056 * A335058 A335059 A335060

KEYWORD

nonn

AUTHOR

Lars Blomberg, May 23 2020

STATUS

approved

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Last modified June 20 23:34 EDT 2021. Contains 345313 sequences. (Running on oeis4.)