A236336 - OEIS

%I #20 Apr 28 2019 18:25:10

%S 1,2,4,5,9,12,16,22,26,33,38,45,53,60,61,76,86,91,92,97,111,112,121,

%T 134,135,147,148,150,153,157,167,180,200,212,223,227,228,238,246,264,

%U 269,282,286,305,312,313,321,322,327,328,360,374,389,393,395,420,421

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

%C An increasing version of A236335.

%H Alois P. Heinz, <a href="/A236336/b236336.txt">Table of n, a(n) for n = 1..10000</a>

%e 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 a:= proc(n) option remember; local i, j, k, ok;

%p       if n<3 then n

%p     else for k from 1+a(n-1) do ok:=true;

%p            for j from n-1 to 2 by -1 while ok do

%p              for i from j-1 to 1 by -1 while ok do

%p                ok:= (n-j)*(a(j)-a(i))<>(j-i)*(k-a(j)) od

%p            od; if ok then return k fi

%p          od

%p       fi

%p     end:

%p seq(a(n), n=1..70);  # _Alois P. Heinz_, Jan 23 2014

%t g[1] = 1;

%t g[n_] := g[n] =

%t   Min[Complement[Range[g[n - 1] + 1, 500],

%t     Select[Flatten[

%t       Table[g[k] + (n - k) (g[j] - g[k])/(j - k), {k, n - 2}, {j,

%t         k + 1, n - 1}]], IntegerQ[#] &]]]

%t Table[g[k], {k, 50}]

%Y Cf. A229037, A185256, A236335.

%K nonn

%O 1,2

%A _Tanya Khovanova_, Jan 22 2014