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

1,1

References

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

Links

Table of n, a(n) for n=1..60.

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.

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.

Example

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}}

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]

Crossrefs

Cf. A046063.

Sequence in context: A201293 A335810 A225016 * A201223 A195721 A021262

Adjacent sequences: A121989 A121990 A121991 * A121993 A121994 A121995

Keyword

nonn,tabf,uned

Author

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

Status

approved