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A286091 Lexicographically earliest sequence of positive integers such that the slope between any two points (i, a(i)) and (j, a(j)) is distinct.

%I A286091 #27 Mar 11 2025 02:55:21
%S A286091 1,1,2,5,11,4,3,18,26,35,48,15,76,64,97,135,29,6,175,98,212,240,260,
%T A286091 73,22,316,41,232,7,165,424,472,399,519,214,353,606,27,660,100,787,
%U A286091 845,924,963,376,156,1095,766,356,621,1206,32,501,1292,1409,1169,1464
%N A286091 Lexicographically earliest sequence of positive integers such that the slope between any two points (i, a(i)) and (j, a(j)) is distinct.
%C A286091 a(1) = a(2) = 1 appears twice; it is the only term that can appear more than once.
%C A286091 From _Peter Kagey_, May 02 2017: (Start)
%C A286091 Sequence is bounded above by (n-1)*A000217(n-2)+1.
%C A286091 Proof: The first n terms of the sequences have (1+2+...+(n-1)) = A000217(n-2) slopes, thus all of the lines starting at any of the first (n - 1) points with any of the already-present slopes can at most cross (n, 1), (n, 2), ... (n, (n-1)*A000217(n-2)). (End)
%H A286091 Peter Kagey and David A. Corneth, <a href="/A286091/b286091.txt">Table of n, a(n) for n = 1..1000 (first 600 terms from Peter Kagey)</a>
%H A286091 David A. Corneth, <a href="/A286091/a286091_1.gp.txt">PARI program</a>
%e A286091 a(3) != 1 otherwise the slope(a(1),a(2)) = slope(a(1),a(3)), therefore
%e A286091 a(3) = 2.
%e A286091 a(4) != 1 otherwise the slope(a(1),a(2)) = slope(a(1),a(4)),
%e A286091 a(4) != 2 otherwise the slope(a(1),a(2)) = slope(a(3),a(4)),
%e A286091 a(4) != 3 otherwise the slope(a(2),a(3)) = slope(a(3),a(4)),
%e A286091 a(4) != 4 otherwise the slope(a(2),a(3)) = slope(a(1),a(4)), therefore
%e A286091 a(4) = 5.
%p A286091 A[1]:= 1:
%p A286091 Slopes:= {}:
%p A286091 for n from 2 to 100 do
%p A286091   for k from 1 do
%p A286091     Sk:= {seq((k-A[i])/(n-i),i=1..n-1)};
%p A286091     if Sk intersect Slopes = {} then
%p A286091       A[n]:= k; Slopes:= Slopes union Sk; break
%p A286091     fi
%p A286091 od od:
%p A286091 seq(A[n],n=1..100); # _Robert Israel_, May 01 2017
%o A286091 (PARI) \\ See link "PARI program". _David A. Corneth_, May 05 2017
%Y A286091 Cf. A236335.
%K A286091 nonn
%O A286091 1,3
%A A286091 _Peter Kagey_, May 01 2017