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A226164 Sequence used for the quadratic irrational number belonging to the principal indefinite binary quadratic form.

%I A226164 #13 Jun 09 2025 01:05:18
%S A226164 1,2,2,3,3,4,3,4,4,5,4,5,5,6,5,6,5,6,6,7,6,7,6,7,7,8,7,8,7,8,7,8,8,9,
%T A226164 8,9,8,9,8,9,9,10,9,10,9,10,9,10,9,10,10,11,10,11,10,11,10,11,10,11,
%U A226164 11,12,11,12,11,12,11,12,11,12,11,12,12,13,12,13,12,13,12,13,12
%N A226164 Sequence used for the quadratic irrational number belonging to the principal indefinite binary quadratic form.
%C A226164 For an indefinite binary quadratic form, denoted by [a, b, c] for F = F([a, b, c],[x, y]) = a*x^2 + b*x*y + c*y^2, the discriminant is D = b^2 - 4*a*c > 0, not a square. See A079896 for the possible values.
%C A226164 The principal form for a discriminant D, which is reduced (see the Scholz-Schoeneberg reference, p. 112), is defined as the unique form F_p(D) = [a=1, b(D), c(D)] with c(D) = -(D - b^2)/4. See the Buell reference, p. 26. One can show that b(D) = f(D) - 2 if D and f(D):=ceiling(sqrt(D(n))) have the same parity and b(D) = f(D) - 1 if D and f(D) have opposite parity. The principal root of a form [a, b, c] of discriminant D is omega(D) = (-b + sqrt(D))/2, the zero with positive square root of the polynomial P(x) = a*x^2 + b*x + c. See the Buell reference, p. 31 (and p. 18). We prefer to call omega the quadratic irrational belonging to the form F. For the principal form F_p(D) of discriminant D = D(n) = A079896(n), n >= 1, this quadratic irrational is omega_p(D(n)) = (-b(D(n)) + sqrt(D))/2 where b(D(n)) is the present sequence a(n). (Note that this differs from the omega = omega(D) used in the Buell reference on p. 40 because another form of discriminant D has been chosen there, depending on the parity of D.)
%C A226164 The (purely periodic) continued fraction expansion of omega_p(D(n)) plays a role for finding all solutions of the Pell equation x^2 + D(n)*y^2 = - 4 if a solution exists. See A226696 for these D values. For the Pell +4 equation which has solutions for every D(n) one finds the fundamental solution also from the continued fraction expansion of omega_p(D(n)).
%C A226164 For more details see the W. Lang link "Periods of indefinite Binary Quadratic Forms ..." given in A225953.
%D A226164 D. A. Buell, Binary Quadratic Forms, Springer, 1989.
%D A226164 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, Sammlung Goeschen Band 5131, Walter de Gruyter, 1973.
%H A226164 Robin Visser, <a href="/A226164/b226164.txt">Table of n, a(n) for n = 1..10000</a>
%F A226164 Define D(n) := A079896(n) and f(n) = ceiling(sqrt(D(n))).
%F A226164 a(n) = f(n) - 2 if D(n) and f(n) have the same parity, and a(n) = f(n) - 1 if D(n) and f(n) have opposite parity.
%e A226164 a(1) = 1 because D(1) = A079896(1) = 5 and f(1) = 3; both are odd, therefore a(1) = 3 - 2 = 1.
%e A226164 a(2) = 2 from D(2) = 8, f(2) = 3, a(2) = f(2) - 1 = 2.
%e A226164 The quadratic irrational (principal root) of the principal form of discriminant D(5) = 17 which is F_p(17) = [1, 3, -2], is omega_p(17) = (-3 + sqrt(17))/2 approximately 0.561552813.
%e A226164   f(17) = 5, a(5) = 5 - 2 = 3 = b(17).
%o A226164 (SageMath)
%o A226164 def a(n):
%o A226164     i, D = 1, Integer(5)
%o A226164     while(i < n):
%o A226164         D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
%o A226164     return ceil(sqrt(D))-1-1*(D%2==ceil(sqrt(D))%2)  # _Robin Visser_, Jun 07 2025
%Y A226164 Cf. A079896, A226696, A225953.
%K A226164 nonn,easy
%O A226164 1,2
%A A226164 _Wolfdieter Lang_, Jul 20 2013
%E A226164 Offset corrected by _Robin Visser_, Jun 07 2025