

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
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OFFSET

1,3


COMMENTS

a(1) = a(2) = 1 appears twice; it is the only term that can appear more than once.
Proof: The first n terms of the sequences have (1+2+...+(n1)) = A000217(n2) slopes, thus all of the lines starting at any of the first (n  1) points with any of the alreadypresent slopes can at most cross (n, 1), (n, 2), ... (n, (n1*A000217(n2)).
(End)


LINKS



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((kA[i])/(ni), i=1..n1)};
if Sk intersect Slopes = {} then
A[n]:= k; Slopes:= Slopes union Sk; break
fi
od od:


PROG



CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



