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A006374
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Number of positive definite reduced binary quadratic forms of discriminant -4*n.
(Formerly M0214)
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5
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1, 1, 2, 2, 2, 2, 2, 3, 3, 2, 4, 4, 2, 4, 4, 4, 4, 3, 4, 6, 4, 2, 6, 6, 3, 6, 6, 4, 6, 4, 6, 7, 4, 4, 8, 8, 2, 6, 8, 6, 8, 4, 4, 10, 6, 4, 10, 8, 5, 7, 8, 6, 6, 8, 8, 12, 4, 2, 12, 8, 6, 8, 10, 8, 8, 8, 4, 12, 8, 4, 14, 9, 4, 10, 10, 10, 8, 4, 10, 14, 9, 4, 12, 12, 4, 10, 12, 6, 12, 10, 8
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OFFSET
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1,3
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COMMENTS
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In Hurwitz and Kritikos (HK) a definite form a*x^2 + 2*b*x*y + c*y^2, denoted by f(a,b,c), has discriminant Delta = -D = b^2 - 4*a*c < 0. Usually this is F = [a,2*b,c] with discriminant Disc = 4*(b^2 - a*c) = 4*Delta. A definite form is reduced if 2*|b| <= a <= c, and if any of the inequalities reduces to an equality then b >= 0 (HK, p. 179). The positive definite case has a > 0 and c > 0. Here the forms F do not need to satisfy gcd(a,2*b,c) = 1. - Wolfdieter Lang, Mar 31 2019
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 360.
A. Hurwitz and N. Kritikos, transl. with add. material by W . C. Schulz, "Lectures on Number Theory", Springer-Verlag, New York, 1986, p. 186.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Table of n, a(n) for n=1..91.
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EXAMPLE
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a(5) = 2 because the two forms F = [a,2*b,c] with discriminant Disc = -4*5 = -20 are [1,0,5] and [2,2,3]. ([2,-2,3] is not reduced, (-2,2,-3) is not positive definite). - Wolfdieter Lang, Mar 31 2019
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CROSSREFS
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Cf. A006371, A006375, A096446, A096445.
Sequence in context: A229942 A025422 A078640 * A193677 A281855 A137921
Adjacent sequences: A006371 A006372 A006373 * A006375 A006376 A006377
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Name clarified by Wolfdieter Lang, Mar 31 2019
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STATUS
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approved
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