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

1,3

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. - Peter Kagey, May 02 2017

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)

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

Example

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.

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

Prog

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

Crossrefs

Cf. A236335.

Sequence in context: A175310 A175311 A246208 * A363554 A127011 A170868

Adjacent sequences: A286088 A286089 A286090 * A286092 A286093 A286094

Keyword

nonn

Author

Peter Kagey, May 01 2017

Status

approved