

A284918


Lexicographically earliest sequence of positive integers such that no three distinct points (i, a(i)), (j, a(j)), (k, a(k)) form an isosceles triangle.


1



1, 1, 2, 2, 6, 4, 8, 8, 12, 13, 5, 12, 16, 16, 17, 21, 11, 17, 27, 7, 13, 27, 7, 3, 3, 10, 29, 20, 35, 19, 34, 28, 28, 35, 20, 9, 41, 42, 46, 30, 22, 44, 25, 31, 32, 51, 54, 58, 47, 49, 61, 29, 63, 61, 19, 47, 45, 71, 39, 25, 69, 67, 71, 74, 53, 33, 85, 72, 81
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OFFSET

1,3


COMMENTS

Degenerate isosceles triangles (i.e., three evenlyspaced points on a line) are also disallowed; otherwise this sequence would be the ones sequence.
For all n != m, if a(n) = a(m) then n  m is odd.
Conjecture: each integer appears exactly twice in this sequence.


LINKS



EXAMPLE

Let p_n = (n, a(n)).
For n = 5, a(5) = 6 because
if a(5) = 1 then (p_1, p_3, p_5) forms an isosceles triangle,
if a(5) = 2 then (p_3, p_4, p_5) forms a degenerate isosceles triangle,
if a(5) = 3 then (p_1, p_3, p_5) forms a degenerate isosceles triangle,
if a(5) = 4 then (p_2, p_4, p_5) forms an isosceles triangle, and
if a(5) = 5 then (p_1, p_4, p_5) forms an isosceles triangle, therefore
a(5) = 6, the least value that does not form an isosceles triangle.


MATHEMATICA

d[p_, q_] := Total[(pq)^2]; bad[a_, b_, c_] := Length[Union[{d[a, b], d[a, c], d[b, c]}]] < 3; good[w_] := Catch[ Do[ If[ bad[ w, L[[i]], L[[j]]], Throw@ False], {i, Length[L]}, {j, i1}]; True]; L = {}; n = 0; While[n < 69, n++; k = 1; While[! good[{n, k}], k++]; AppendTo[L, {n, k}]]; Last /@ L (* Giovanni Resta, Apr 06 2017 *)


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KEYWORD



AUTHOR



STATUS

approved



