A201223 - OEIS

%I #30 Mar 22 2015 10:06:34

%S 3,8,7,5,8,7,7,15,13,8,15,13,5,21,19,16,21,19,11,35,31,24,35,31,7,40,

%T 37,33,40,37,13,48,43,35,48,43,16,55,49,39,55,49,9,65,61,56,65,61,32,

%U 77,67,45,77,67,17,80,73,63,80,73,40,91,79,51,91,79,11,96,91

%N Primitive Eisenstein triples (a,b,c) listed as groups of three in order of increasing b.

%C An Eisenstein triple is a triple (a,b,c) of positive integers with a<c<b and a^2 - a*b + b^2 = c^2.  It is primitive if a, b, and c are relatively prime.

%H Danny Rorabaugh, <a href="/A201223/b201223.txt">Table of n, a(n) for n = 1..1494</a>

%H Russell A. Gordon, <a href="http://www.jstor.org/stable/10.4169/math.mag.85.1.12">Properties of Eisenstein Triples</a>, Mathematics Magazine 85 (2012), 12-25.

%e (a,b,c)=(3,8,7) is an Eisenstein triple since 3<7<8 and 3^2 - 3*8 + 8^2 = 7^2.  GCD(3,8,7) = 1, so the triple is primitive.  No Eisenstein triple exists with b<8, so a(1)=3, a(2)=8, a(3)=7.

%t x = {}; For[b = 1, b <= 77, b++, For[c = 1, c < b, c++, For[a = 1, a < c, a++, {If[(a^2 - a*b + b^2 == c^2) && (GCD[a, b, c] == 1), AppendTo[x, {a, b, c}]]}]]]; Flatten[x]

%Y Cf. A121992.

%K nonn

%O 1,1

%A _John W. Layman_, May 09 2012