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A231334 Lexicographically earliest sequence of distinct positive integers such that for any distinct i,j, k, the points at positions (i, a(i)), (j, a(j)), (k, a(k)) are not aligned.

%I A231334 #22 Apr 19 2020 07:51:04
%S A231334 1,2,4,3,6,5,9,12,7,14,13,8,23,17,18,22,10,15,11,28,19,16,20,29,32,44,
%T A231334 35,39,24,40,26,37,42,21,56,64,43,31,25,34,27,33,66,67,52,60,30,57,36,
%U A231334 63,86,82,38,50,47,69,75,79,89,49,45,76,41,48,98,77,94
%N A231334 Lexicographically earliest sequence of distinct positive integers such that for any distinct i,j, k, the points at positions (i, a(i)), (j, a(j)), (k, a(k)) are not aligned.
%C A231334 Is this a permutation of the natural numbers?
%C A231334 There are only two fixed points: 1 and 2.
%H A231334 Paul Tek, <a href="/A231334/b231334.txt">Table of n, a(n) for n = 1..10000</a>
%H A231334 Paul Tek, <a href="/A231334/a231334.txt">C program for this sequence</a>
%H A231334 Illustrations of the first 200 points at positions (n, a(n)) (black pixels correspond to these points, colored pixels (x,y) are aligned with two black pixels (i,a(i)) and (j,a(j))):
%H A231334 (1) <a href="/A231334/a231334.png">x < i < j</a>,
%H A231334 (2) <a href="/A231334/a231334_1.png">i < x < j</a>,
%H A231334 (3) <a href="/A231334/a231334_2.png">i < j < x</a>.
%t A231334 WIDTH = 1000;
%t A231334 HEIGHT = 2000;
%t A231334 Clear[seen, aligned, a];
%t A231334 compute[n_] := Module[{c = 1}, While[seen[c] || aligned[n][c], c++; If[c > HEIGHT, Abort[]]]; a[n] = c; seen[a[n]] = True; For[i = 1, i < n, i++, dn = n - i; da = a[n] - a[i]; g = GCD[dn, da]; dn /= g; da /= g; nn = n; na = c; While[True, nn += dn; If[nn > WIDTH, Break[]]; na += da; If[na < 1 || na > HEIGHT, Break[]]; aligned[nn][na] = True]]; a[n]];
%t A231334 Array[compute, WIDTH] (* _Jean-François Alcover_, Apr 19 2020, translated from _Paul Tek_'s program. *)
%o A231334 (C) See Link section.
%Y A231334 Cf. A175498, A236266, A236335, A300002.
%K A231334 nonn
%O A231334 1,2
%A A231334 _Paul Tek_, Nov 07 2013