A236266 Lexicographically earliest sequence of nonnegative integers such that no three points (i,a(i)), (j,a(j)), (n,a(n)) are collinear.

0, 0, 1, 1, 4, 3, 8, 2, 2, 5, 7, 4, 5, 8, 16, 3, 7, 14, 12, 23, 16, 12, 25, 31, 13, 6, 11, 28, 11, 17, 9, 9, 22, 34, 6, 15, 13, 29, 23, 22, 29, 45, 26, 19, 51, 14, 24, 39, 28, 39, 18, 37, 57, 17, 38, 41, 15, 68, 32, 24, 66, 42, 10, 50, 27, 10, 53, 72, 25, 26

Offset: 0

Author

Alois P. Heinz, Jan 21 2014

Keywords

• nonn
• look

Comments

(a(n)-a(j))/(n-j) <> (a(j)-a(i))/(j-i) for all 0<=i

Examples

			For n=4 the value of a(n) cannot be less than 4 because otherwise we would have a set of three collinear points, {(0,0),(1,0),(4,0)} or {(2,1),(3,1),(4,1)} or {(0,0),(2,1),(4,2)} or {(1,0),(2,1),(4,3)}.  Thus a(4) = 4 is the first value that is in accordance with the constraints.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000
• Dániel T. Nagy, Zoltán Lóránt Nagy, and Russ Woodroofe, The extensible No-Three-In-Line problem, arXiv:2209.01447 [math.CO], 2022.

Crossrefs

Cf. A005836, A179040, A231334, A236335, A255708, A255709.

Programs

Maple

a:= proc(n) option remember; local i, j, k, ok;
      for k from 0 do ok:=true;
        for j from n-1 to 1 by -1 while ok do
          for i from j-1 to 0 by -1 while ok do
            ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j)) od
        od; if ok then return k fi
      od
    end:
seq(a(n), n=0..60);

Mathematica

a[0] = a[1] = 0; a[n_] := a[n] = Module[{i, j, k, ok}, For[k = 0, True, k++, ok = True; For[j = n-1, ok && j >= 1, j--, For[i = j-1, ok && i >= 0, i--, ok = (n-j)*(a[j]-a[i]) != (j-i)*(k-a[j])]]; If[ok, Return[k]]]];
Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Jun 16 2018, after Alois P. Heinz *)

Formula

a(n) = A236335(n+1) - 1. - Alois P. Heinz, Jan 23 2014

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