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

1, 1, 2, 5, 11, 4, 3, 18, 26, 35, 48, 15, 76, 64, 97, 135, 29, 6, 175, 98, 212, 240, 260, 73, 22, 316, 41, 232, 7, 165, 424, 472, 399, 519, 214, 353, 606, 27, 660, 100, 787, 845, 924, 963, 376, 156, 1095, 766, 356, 621, 1206, 32, 501, 1292, 1409, 1169, 1464

Offset: 1

Author

Peter Kagey, May 01 2017

Keywords

• nonn

Comments

a(1) = a(2) = 1 appears twice; it is the only term that can appear more than once.

From Peter Kagey, May 02 2017: (Start)

Sequence is bounded above by (n-1)*A000217(n-2)+1.

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)

Examples

			a(3) != 1 otherwise the slope(a(1),a(2)) = slope(a(1),a(3)), therefore
a(3) = 2.
a(4) != 1 otherwise the slope(a(1),a(2)) = slope(a(1),a(4)),
a(4) != 2 otherwise the slope(a(1),a(2)) = slope(a(3),a(4)),
a(4) != 3 otherwise the slope(a(2),a(3)) = slope(a(3),a(4)),
a(4) != 4 otherwise the slope(a(2),a(3)) = slope(a(1),a(4)), therefore
a(4) = 5.
		

Links

• Peter Kagey and David A. Corneth, Table of n, a(n) for n = 1..1000 (first 600 terms from Peter Kagey)
• David A. Corneth, PARI program

Crossrefs

Cf. A236335.

Programs

Maple

A[1]:= 1:
Slopes:= {}:
for n from 2 to 100 do
  for k from 1 do
    Sk:= {seq((k-A[i])/(n-i),i=1..n-1)};
    if Sk intersect Slopes = {} then
      A[n]:= k; Slopes:= Slopes union Sk; break
    fi
od od:
seq(A[n],n=1..100); # Robert Israel, May 01 2017

PARI

\\ See link "PARI program". David A. Corneth, May 05 2017