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
KEYWORD
nonn,look
AUTHOR
Alois P. Heinz, Jan 21 2014
STATUS
approved