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A257008
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Number of Zagier-reduced binary quadratic forms of discriminant n^2+4.
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2
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1, 2, 3, 5, 5, 10, 7, 13, 14, 16, 12, 31, 13, 24, 29, 38, 17, 44, 26, 47, 46, 34, 30, 90, 34, 56, 49, 63, 39, 106, 40, 87, 77, 70, 57, 139, 55, 58, 89, 149, 52, 138, 52, 136, 123, 92, 69, 223, 84, 104, 146, 111, 62, 218, 94, 214, 121, 132, 96, 296
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OFFSET
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1,2
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COMMENTS
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The number of finite sequences of positive integers with even length parity and alternant equal to n.
The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=n^2+4.
The number of possible asymmetry types for the quotient sequence of the odd-length continued fraction expansion of a rational number a/b, where b satisfies one of the congruences b^2 + nb - 1 = 0 (mod a) or b^2 - nb - 1 = 0 (mod a)
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REFERENCES
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D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.
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LINKS
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FORMULA
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With D=n^2+4, a(n) equals the number of pairs (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4.
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EXAMPLE
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For n=4, the a(4) = 5 Zagier-reduced forms of discriminant 20 are x^2 + 6*x*y + 4*y^2, 4*x^2 + 6*x*y + y^2, 4*x^2 + 10*x*y + 5*y^2, 5*x^2 + 10*x*y + 4*y^2, and 2*x^2 + 6*x*y + 2*y^2
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MATHEMATICA
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Table[Length[
Flatten[
Select[
Table[{a, k}, {k,
Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
Mod[# - n, 2] == 0 &]}, {a,
Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
UnsameQ[#, {}] &], 1]], {n, Map[#^2 + 4 &, Range[3, 60]]}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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