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

Author

Tanya Khovanova, Jan 22 2014

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

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

Programs

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}]

Formula

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