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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

Table of n, a(n) for n=1..60.

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]<b then t2:=t2+1; fi; od;

t2; end;

f:=n->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 February 19 10:03 EST 2020. Contains 332041 sequences. (Running on oeis4.)