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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Degenerate isosceles triangles (i.e., three evenly-spaced 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

Peter Kagey, Table of n, a(n) for n = 1..500

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[(p-q)^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, i-1}]; 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 *)

CROSSREFS

Sequence in context: A071059 A256468 A061108 * A053213 A292258 A140524

Adjacent sequences:  A284915 A284916 A284917 * A284919 A284920 A284921

KEYWORD

nonn,look

AUTHOR

Peter Kagey, Apr 05 2017

STATUS

approved

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Last modified November 12 19:19 EST 2018. Contains 317116 sequences. (Running on oeis4.)