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.

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, 35, 39, 24, 40, 26, 37, 42, 21, 56, 64, 43, 31, 25, 34, 27, 33, 66, 67, 52, 60, 30, 57, 36, 63, 86, 82, 38, 50, 47, 69, 75, 79, 89, 49, 45, 76, 41, 48, 98, 77, 94

Offset: 1

Author

Paul Tek, Nov 07 2013

Keywords

• nonn

Comments

Is this a permutation of the natural numbers?

There are only two fixed points: 1 and 2.

Links

• Paul Tek, Table of n, a(n) for n = 1..10000
• Paul Tek, C program for this sequence
• 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))):
• (1) x < i < j,
• (2) i < x < j,
• (3) i < j < x.

Crossrefs

Cf. A175498, A236266, A236335, A300002.

Programs

C

See Link section.

Mathematica

WIDTH = 1000;
HEIGHT = 2000;
Clear[seen, aligned, a];
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]];
Array[compute, WIDTH] (* Jean-François Alcover, Apr 19 2020, translated from Paul Tek's program. *)