A284918 - OEIS

%I #22 Jun 29 2017 11:47:08

%S 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,

%T 20,35,19,34,28,28,35,20,9,41,42,46,30,22,44,25,31,32,51,54,58,47,49,

%U 61,29,63,61,19,47,45,71,39,25,69,67,71,74,53,33,85,72,81

%N 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.

%C Degenerate isosceles triangles (i.e., three evenly-spaced points on a line) are also disallowed; otherwise this sequence would be the ones sequence.

%C For all n != m, if a(n) = a(m) then n - m is odd.

%C Conjecture: each integer appears exactly twice in this sequence.

%H Peter Kagey, <a href="/A284918/b284918.txt">Table of n, a(n) for n = 1..500</a>

%e Let p_n = (n, a(n)).

%e For n = 5, a(5) = 6 because

%e if a(5) = 1 then (p_1, p_3, p_5) forms an isosceles triangle,

%e if a(5) = 2 then (p_3, p_4, p_5) forms a degenerate isosceles triangle,

%e if a(5) = 3 then (p_1, p_3, p_5) forms a degenerate isosceles triangle,

%e if a(5) = 4 then (p_2, p_4, p_5) forms an isosceles triangle, and

%e if a(5) = 5 then (p_1, p_4, p_5) forms an isosceles triangle, therefore

%e a(5) = 6, the least value that does not form an isosceles triangle.

%t 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 *)

%K nonn,look

%O 1,3

%A _Peter Kagey_, Apr 05 2017