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A236336 Lexicographically earliest increasing sequence of positive integers whose graph has no three collinear points.

%I A236336 #20 Apr 28 2019 18:25:10
%S A236336 1,2,4,5,9,12,16,22,26,33,38,45,53,60,61,76,86,91,92,97,111,112,121,
%T A236336 134,135,147,148,150,153,157,167,180,200,212,223,227,228,238,246,264,
%U A236336 269,282,286,305,312,313,321,322,327,328,360,374,389,393,395,420,421
%N A236336 Lexicographically earliest increasing sequence of positive integers whose graph has no three collinear points.
%C A236336 An increasing version of A236335.
%H A236336 Alois P. Heinz, <a href="/A236336/b236336.txt">Table of n, a(n) for n = 1..10000</a>
%e A236336 Consider a(5). The previous terms are 1,2,4,5. The value of a(5) can't be 6 because points (3,4),(4,5),(5,6) (corresponding to values a(3),a(4),a(5)) are on the same line: y=x+1. Points (1,1),(3,4),(5,7) are on the same line y=3x/2-1/2, so a(5) can't be 7. Points (2,2),(3,4),(5,8) are on the same line: y=2x-2, so a(5) can't be 8. Thus a(5)=5.
%p A236336 a:= proc(n) option remember; local i, j, k, ok;
%p A236336       if n<3 then n
%p A236336     else for k from 1+a(n-1) do ok:=true;
%p A236336            for j from n-1 to 2 by -1 while ok do
%p A236336              for i from j-1 to 1 by -1 while ok do
%p A236336                ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j)) od
%p A236336            od; if ok then return k fi
%p A236336          od
%p A236336       fi
%p A236336     end:
%p A236336 seq(a(n), n=1..70);  # _Alois P. Heinz_, Jan 23 2014
%t A236336 g[1] = 1;
%t A236336 g[n_] := g[n] =
%t A236336   Min[Complement[Range[g[n - 1] + 1, 500],
%t A236336     Select[Flatten[
%t A236336       Table[g[k] + (n - k) (g[j] - g[k])/(j - k), {k, n - 2}, {j,
%t A236336         k + 1, n - 1}]], IntegerQ[#] &]]]
%t A236336 Table[g[k], {k, 50}]
%Y A236336 Cf. A229037, A185256, A236335.
%K A236336 nonn
%O A236336 1,2
%A A236336 _Tanya Khovanova_, Jan 22 2014