A236335 Lexicographically earliest sequence of positive integers whose graph has no three collinear points.

1, 1, 2, 2, 5, 4, 9, 3, 3, 6, 8, 5, 6, 9, 17, 4, 8, 15, 13, 24, 17, 13, 26, 32, 14, 7, 12, 29, 12, 18, 10, 10, 23, 35, 7, 16, 14, 30, 24, 23, 30, 46, 27, 20, 52, 15, 25, 40, 29, 40, 19, 38, 58, 18, 39, 42, 16, 69, 33, 25, 67, 43, 11, 51, 28, 11, 54, 73, 26, 27

Offset

1,3

Comments

An integer can't appear more than twice in the sequence, which means the sequence tends to infinity.

An increasing version of this sequence is A236336.

Links

Grant Garcia, Table of n, a(n) for n = 1..10000

Dániel T. Nagy, Zoltán Lóránt Nagy, and Russ Woodroofe, The extensible No-Three-In-Line problem, arXiv:2209.01447 [math.CO], 2022.

Formula

a(n) = A236266(n-1) + 1. - Alois P. Heinz, Jan 23 2014

Example

Consider a(5). The previous terms are 1,1,2,2. The value of a(5) can't be 1 because points (1,1),(2,1),(5,1) (corresponding to values a(1), a(2), a(5)) are on the same line: y=1. Points (3,2),(4,2),(5,2) are on the same line y=2, so a(5) can't be 2. Points (1,1),(3,2),(5,3) are on the same line: y=x/2+1/2, so a(5) can't be 3. Points (2,1),(3,2),(5,4) are on the same line: y=x-1, so a(5) can't be 4. Thus a(5)=5.

Mathematica

b[1] = 1;
b[n_] := b[n] =
  Min[Complement[Range[100],
    Select[Flatten[
      Table[b[k] + (n - k) (b[j] - b[k])/(j - k), {k, n - 2}, {j,
        k + 1, n - 1}]], IntegerQ[#] &]]]
Table[b[k], {k, 70}]

Crossrefs

Cf. A229037, A185256, A231334, A236266, A236336, A300002.

Sequence in context: A034400 A021820 A222882 * A016724 A058657 A321285

Adjacent sequences: A236332 A236333 A236334 * A236336 A236337 A236338

Keyword

nonn

Author

Tanya Khovanova, Jan 22 2014

Status

approved