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.

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

Offset: 1

Author

Roger L. Bagula and Gary W. Adamson, Sep 10 2006

Keywords

• nonn
• tabf
• uned

Examples

			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}}
		

References

• Ross Honsberger, "Mathematical Delights", MAA, 2004, p. 64.

Links

• Park City Mathematics Institute, Session 13 Number Theory, 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.

Crossrefs

Cf. A046063.

Programs

Mathematica

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]

Formula

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.