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 A257007 Number of Zagier-reduced binary quadratic forms of discriminant n^2-4. 5
 0, 0, 1, 3, 4, 7, 7, 12, 8, 20, 13, 18, 18, 31, 20, 31, 24, 39, 26, 53, 20, 66, 36, 36, 50, 76, 39, 62, 56, 92, 42, 72, 42, 120, 68, 72, 70, 136, 46, 126, 76, 112, 100, 96, 68, 146, 105, 125, 66, 226, 77, 168, 96, 138, 126, 160, 96, 228, 100, 142 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The number of finite sequences of positive integers with odd 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 even-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) REFERENCES D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981. LINKS P. Kleban and A. Özlük, A farey fraction spin chain, Communications in mathematical physics, 203(3):635-647, 1999. This sequence appears to be the function Phi(n) given in Theorem 4. B. R. Smith, Reducing quadratic forms by kneading sequences J. Int. Seq., 17 (2014) 14.11.8. B. R. Smith, End-symmetric continued fractions and quadratic congruencesActa Arith., 167 (2015) 173-187. M. Technau, The Calkin-Wilf tree and a trace condition, Master's Thesis, 2015. The sequence appears to be the function N(n,0) from subsection 1.3.1. FORMULA 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. From the Kleban et al. reference it appears that a(n) = Sum_{b=1..n-1} dbm(b,n*b-b^2-1), where dbm(b,m) = number of positive divisors of m that are less than b. - N. J. A. Sloane, Nov 19 2015 EXAMPLE For n=5, the a(5) = 4 Zagier-reduced forms of discriminant 21 are x^2 + 5*x*y + y^2, 5*x^2 + 9*x*y + 3*y^2, 3*x^2 + 9*x*y + 5*y^2, and 5*x^2 + 11*x*y + 5*y^2. MAPLE # Maple code for the formula given by Kleban et al., which is almost certainly the same sequence as this (but until that is proved, the program should not be used to extend this sequence, A264598 or A264599). - N. J. A. Sloane, Nov 19 2015 with(numtheory); # return number of divisors of m less than b dbm:=proc(b, m) local i, t1, t2; t1:=divisors(m); t2:=0; for i from 1 to nops(t1) do if t1[i]add(dbm(b, b*n-b^2-1), b=1..n-1); [seq(f(n), n=1..100)]; MATHEMATICA 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]]}] CROSSREFS Cf. A257003, A257008, A257009. It appears that this sequence gives half the row sums of the triangle in A264597 (cf. A264598), and also the first column of A264597. - N. J. A. Sloane, Nov 19 2015 Sequence in context: A168563 A261368 A307743 * A070324 A054155 A054058 Adjacent sequences:  A257004 A257005 A257006 * A257008 A257009 A257010 KEYWORD nonn AUTHOR Barry R. Smith, Apr 16 2015 STATUS approved

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Last modified October 1 01:28 EDT 2022. Contains 357134 sequences. (Running on oeis4.)