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A121992 List of Eisenstein triples: {a,b,c} such that {a^2 + b^2 - a*b - c^2 = 0} and abs(a - b) > 0, sorted by greatest a.

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%I A121992 #19 Jan 30 2021 01:25:22
%S A121992 3,8,7,5,8,7,5,21,19,6,16,14,7,15,13,8,3,7,8,5,7,8,15,13,9,24,21,10,
%T A121992 16,14,15,7,13,15,8,13,15,24,21,16,6,14,16,10,14,16,21,19,21,5,19,21,
%U A121992 16,19,24,9,21,24,15,21
%N A121992 List of Eisenstein triples: {a,b,c} such that {a^2 + b^2 - a*b - c^2 = 0} and abs(a - b) > 0, sorted by greatest a.
%D A121992 Ross Honsberger, "Mathematical Delights", MAA, 2004, p. 64.
%H A121992 Park City Mathematics Institute, <a href="http://mathforum.org/pcmi/hstp/sum2001/wg/number.theory/session13.pdf">Session 13 Number Theory</a>, Summer 2001. A similar factoring allows for the generation of Eisenstein triples, which are numbers that form the sides of a triangle with a 60-degree angle.
%F A121992 T(n) = {a(n), b(n), c(n)} such that a(n)^2 + b(n)^2 - a(n)*b(n) - c(n)^2 = 0 and abs(a(n) - b(n)) > 0.
%e A121992 Grouped as threes: {{3, 8, 7}, {5, 8, 7}, {5, 21, 19}, {6, 16, 14}, {7, 15, 13}, {8, 3, 7}, {8, 5, 7}, {8, 15, 13}, {9, 24, 21}, {10, 16, 14}, {15, 7, 13}, {15, 8, 13}, {15, 24, 21}, {16,6, 14}, {16, 10, 14}, {16, 21, 19}, {21, 5, 19}, {21, 16, 19}, {24, 9, 21}, {24, 15, 21}}
%t A121992 f[a_, b_, c_] = If[c^2 - a^2 - b^2 + a*b == 0 && Abs[a - b] > 0, {a, b, c}, {}] a0 = Flatten[Delete[Union[Table[Delete[Union[Table[Flatten[Table[f[a, b, c], {c, 1, 25}]], {b, 1, 25}]], 1], {a, 1, 25}]], 1], 1] b0 = Sort[a0] Flatten[b0]
%Y A121992 Cf. A046063.
%K A121992 nonn,tabf,uned
%O A121992 1,1
%A A121992 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 10 2006