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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn
AUTHOR
Peter Kagey, May 01 2017
STATUS
approved

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Last modified February 25 02:17 EST 2024. Contains 370308 sequences. (Running on oeis4.)