A324250 Sequence a(n) = 3*A002559(n) - 2 determining the principal reduced indefinite binary quadratic form [1, a(n), -a(n)] for Markoff triples.

1, 4, 13, 37, 85, 100, 265, 505, 580, 697, 1297, 1828, 2953, 3973, 4789, 8689, 12541, 17221, 19396, 22681, 27229, 32836, 44101, 85969, 100381, 112996, 129781, 154921, 186628, 225073, 289669, 405409, 585073, 589252, 884053, 1279165, 1498177, 1542685, 1938052, 2777293, 3410065, 3836452, 4038805

Offset: 1

Author

Wolfdieter Lang, Mar 04 2019

Keywords

• nonn
• easy

Comments

The indefinite binary quadratic form F(n,x,y) = x^2 - 3*m(n)*x*y + y^2 = [1, -3*m(n), 1] representing -m(n)^2 with m(n) = A002559(n), determines Markoff triples MT(n) = (x(n) = A305313(n), y(n) = A305314(n), m(n)) with x(n) < y(n) < m(n), for n >= 3. For n = 1 and 2: x(n) = y(n) = 1. The Frobenius-Markoff conjecture is that this solution is unique. This form F(n,x,y) has discriminant D(n) = (3*m(n))^2 - 4 = a(n)*(a(n) + 4) = A305312(n) > 0.

Because -3*m(n) < 0 this form F(n,x,y) is not reduced (see e.g., the Buell reference, or the W. Lang link in A225953 for the definition).

The principal reduced form for F(n,x,y) is prF(n,X,Y) = X^2 + a(n)*X*Y - a(n)*Y^2 = [1, a(n), -a(n)]. (See, e.g., Lemma 2 of the W. Lang link in A225953 where b = a(n), f(D(n)) = ceiling(sqrt(D(n))) = 3*m(n), and D(n) and f(D(n)) have the same parity.) The relation between these forms is F(n,Y,Y-X) = prF(n,X,Y) with Y > 0, Y-X > 0, and X <= 0 (for n >= 3, X < 0).

Examples

			n = 3 with a(3) = 13: MT(3) = (1, 2, 5), F(3,x,y) = [1, -3*5, 1], prF(3,X,Y) = [1, 13, -13]. prF(3,X,Y) = -5^2 has two proper fundamental solutions with Y > 0, namely (-1, 1) and (1, 2). The unique solution with Y > 0, X < 0, and Y-X < 5 is (X, Y) = (-1, 1) corresponding to (x,y) = (1, 2) for MT(3).
The other fundamental solution (1, 2) corresponds to the unordered Markoff triple (2, 1, 5) (x > y, X > 0). The next solution in this class with X < 0 is (-12, 1) corresponding to the unordered triple (1, 13, 5) (Y-X = 13 > 5).
		

References

• D. A. Buell, Binary quadratic forms, 1989, Springer, p. 21.

Crossrefs

Cf. A002559, A305312, A305313, A305314.

Formula

a(n) = 3*A002559(n) - 2, for n >= 1.