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, 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

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