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%I #39 Apr 18 2023 05:27:27
%S 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,
%T 76,39,62,56,92,42,72,42,120,68,72,70,136,46,126,76,112,100,96,68,146,
%U 105,125,66,226,77,168,96,138,126,160,96,228,100,142
%N Number of Zagier-reduced binary quadratic forms of discriminant n^2-4.
%C The number of finite sequences of positive integers with odd length parity and alternant equal to n.
%C 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.
%C 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)
%D D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.
%H P. Kleban and A. Özlük, <a href="http://arxiv.org/abs/cond-mat/9808182">A Farey fraction spin chain</a>, arXiv:cond-mat/9808182 [cond-mat.stat-mech], 1998; Communications in mathematical physics, 203(3):635-647, 1999. This sequence appears to be the function Phi(n) given in Theorem 4.
%H B. R. Smith, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Smith/smith5.html">Reducing quadratic forms by kneading sequences</a> J. Int. Seq., 17 (2014) 14.11.8.
%H B. R. Smith, <a href="https://doi.org/10.4064/aa167-2-5">End-symmetric continued fractions and quadratic congruences</a>, Acta Arith., 167 (2015) 173-187.
%H Marc Technau, <a href="https://web.archive.org/web/20171109080920/http://www.mathematik.uni-wuerzburg.de/~steuding/technau.pdf">The Calkin-Wilf tree and a trace condition</a>, Master's Thesis, 2015. The sequence appears to be the function N(n,0) from subsection 1.3.1.
%H Marc Technau, <a href="https://arxiv.org/abs/2304.08143">Remark on the Farey fraction spin chain</a>, arXiv:2304.08143 [math.NT], 2023. See Theorem 2 p. 4.
%F 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.
%F 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
%e 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.
%p # 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
%p with(numtheory); # return number of divisors of m less than b
%p dbm:=proc(b,m) local i,t1,t2;
%p t1:=divisors(m); t2:=0;
%p for i from 1 to nops(t1) do if t1[i]<b then t2:=t2+1; fi; od;
%p t2; end;
%p f:=n->add(dbm(b,b*n-b^2-1), b=1..n-1);
%p [seq(f(n),n=1..100)];
%t Table[Length[
%t Flatten[
%t Select[
%t Table[{a, k}, {k,
%t Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
%t Mod[# - n, 2] == 0 &]}, {a,
%t Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
%t UnsameQ[#, {}] &], 1]], {n, Map[#^2 - 4 &, Range[3, 60]]}]
%o (PARI) d(n, k) = #select(x->(x<n), divisors(k));
%o a(n) = if (n<=2, 0, sum(i=1, n-1, d(i, i*n-i^2-1))); \\ _Michel Marcus_, Apr 18 2023; based on Technau Lemma 3
%Y Cf. A257003, A257008, A257009.
%Y 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
%K nonn
%O 1,4
%A _Barry R. Smith_, Apr 16 2015
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