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

0,5

Comments

(a(n)-a(j))/(n-j) <> (a(j)-a(i))/(j-i) for all 0<=i<j<n. No value occurs more than twice. Each triangle with (distinct) vertices (i,a(i)), (j,a(j)), (n,a(n)) has area larger than zero.

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.

Formula

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A199077 A345133 A016703 * A099289 A231892 A065628

Adjacent sequences: A236263 A236264 A236265 * A236267 A236268 A236269

Keyword

nonn,look

Author

Alois P. Heinz, Jan 21 2014

Status

approved