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A226164
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Sequence used for the quadratic irrational number belonging to the principal indefinite binary quadratic form.
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0
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1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 4, 5, 5, 6, 5, 6, 5, 6, 6, 7, 6, 7, 6, 7, 7, 8, 7, 8, 7, 8, 7, 8, 8, 9, 8, 9, 8, 9, 8, 9, 9, 10, 9, 10, 9, 10, 9, 10, 9, 10, 10, 11, 10, 11, 10, 11, 10, 11, 10, 11, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 11, 12, 12, 13, 12, 13, 12, 13, 12, 13, 12
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OFFSET
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0,2
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COMMENTS
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For an indefinite binary quadratic form, denoted by [a, b, c] for F = F([a, b, c],[x, y]) = a*x^2 + b*x*y + c*y^2, the discriminant is D = b^2 - 4*a*c > 0, not a square. See A079896 for the possible values.
The principal form for a discriminant D, which is reduced (see the Scholz-Schoeneberg reference, p. 112), is defined as the unique form F_p(D) = [a=1, b(D), c(D)] with c(D) = -(D - b^2)/4. See the Buell reference, p. 26. One can show that b(D) = f(D) - 2 if D and f(D):=ceiling(sqrt(D(n))) have the same parity and b(D) = f(D) - 1 if D and f(D) have opposite parity. The principal root of a form [a, b, c] of discriminant D is omega(D) = (-b + sqrt(D))/2, the zero with positive square root of the polynomial P(x) = a*x^2 + b*x + c. See the Buell reference, p. 31 (and p. 18). We prefer to call omega the quadratic irrational belonging to the form F. For the principal form F_p(D) of discriminant D = D(n) = A079896(n), n >= 0, this quadratic irrational is omega_p(D(n)) = (-b(D(n)) + sqrt(D))/2 where b(D(n)) is the present sequence a(n). (Note that this differs from the omega = omega(D) used in the Buell reference on p. 40 because another form of discriminant D has been chosen there, depending on the parity of D.)
The (purely periodic) continued fraction expansion of omega_p(D(n)) plays a role for finding all solutions of the Pell equation x^2 + D(n)*y^2 = - 4 if a solution exists. See A226696 for these D values. For the Pell +4 equation which has solutions for every D(n) one finds the fundamental solution also from the continued fraction expansion of omega_p(D(n)).
For more details see the W. Lang link "Periods of indefinite Binary Quadratic Forms ..." given in A225953.
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REFERENCES
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D. A. Buell, Binary Quadratic Forms, Springer, 1989.
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Goeschen Band 5131, Walter de Gruyter, 1973.
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LINKS
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Table of n, a(n) for n=0..80.
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FORMULA
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Define D(n) := A079896(n) and f(n) = ceiling(sqrt(D(n))).
a(n) = f(n) - 2 if D(n) and f(n) have the same parity, and a(n) = f(n) - 1 if D(n) and f(n) have opposite parity.
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EXAMPLE
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a(0) = 1 because D(0) = A079896(0) = 5 and f(0) = 3; both are odd, therefore a(0) = 3 - 2 = 1.
a(1) = 2 from D(1) = 8, f(1) = 3, a(1) = f(1) - 1 = 2.
The quadratic irrational (principal root) of the principal form of discriminant D(4) = 17 which is F_p(17) = [1, 3, -2], is omega_p(17) = (-3 + sqrt(17))/2 approximately 0.561552813.
f(17) = 5, a(4) = 5 - 2 = 3 = b(17).
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CROSSREFS
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Cf. A079896, A226696, A225953.
Sequence in context: A064097 A014701 A207034 * A308220 A302039 A056239
Adjacent sequences: A226161 A226162 A226163 * A226165 A226166 A226167
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KEYWORD
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nonn,easy
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AUTHOR
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Wolfdieter Lang, Jul 20 2013
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STATUS
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approved
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