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A324250
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Sequence a(n) = 3*A002559(n) - 2 determining the principal reduced indefinite binary quadratic form [1, a(n), -a(n)] for Markoff triples.
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3
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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
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OFFSET
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1,2
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COMMENTS
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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).
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REFERENCES
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D. A. Buell, Binary quadratic forms, 1989, Springer, p. 21.
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LINKS
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FORMULA
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a(n) = 3*A002559(n) - 2, for n >= 1.
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EXAMPLE
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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).
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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