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%I #19 May 17 2022 02:05:10
%S 1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,1,1,1,2,0,0,1,0,2,0,1,1,2,1,1,0,0,2,0,
%T 0,0,1,1,2,1,0,1,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,1,1,
%U 2,1,2,2,2,0,0,0,2,0
%N Array A(n, k) read by upwards antidiagonals giving the number of representative parallel primitive binary quadratic forms for discriminant Disc(n) = 4*D(n), with D(n) = A000037(n), and for representable integer |k| >= 1.
%C For the definition of representative parallel primitive forms (rpapfs) for discriminant Disc > 0 (the indefinite case) and representation of nonzero integers k see the Scholz-Schoeneberg reference, p. 105, or the Buell reference p. 49 (without use of the name parallel). For the procedure to find the primitive representative parallel forms (rpapfs) for Disc(n) = 4*D(n) = 4*A000037(n) and nonzero integer k see the W. Lang link given in A324251, section 3.
%C Note that the number of rpapfs of a discriminant Disc > 0 for k >= 1 is identical with the one for negative k. These forms differ in the signs of the a and c entries of these forms but not the b >= 0 entry (called an outer sign flip). See some examples below, and the program in the mentioned W. Lang link, section 3.
%C For the forms counted in the array A(n, k) see Table 3 of the W. Lang link given in A324251, for n = 1..30 and k = 1..10.
%C Compare the present array with the ones given in A324252 and A307303 for the number of rpapfs for discriminant 4*D(n) and representable positive and negative k, respectively, that are equivalent (under SL(2, Z)) to the reduced principal form F_p = [1, 2*s(n), -(D(n) - s(n)^2)] with s(n) = A000194(n), of the unreduced Pell form F(n) = [1, 0, -D(n)].
%C The rpapfs not counted in A324252 and A307303 are equivalent to forms of non-principal cycles for discriminant 4*D(n).
%C The total number of cycles (the class number h(n)) for discriminant 4*D(n) is given in A307359(n).
%C The array for the length of the periods of these cycles is given in A307378.
%C One half of the sum of the length of the periods is given in A307236.
%D D. A. Buell, Binary Quadratic Forms, Springer, 1989, chapter 3, pp. 21 - 43.
%D A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, pp. 112 - 126.
%e The array A(n, k) begins:
%e n, D(n) \k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e -------------------------------------------------------------
%e 1, 2: 1 1 0 0 0 0 2 0 0 0 0 0 0 2 0
%e 2, 3: 1 1 1 0 0 1 0 0 0 0 2 0 2 0 0
%e 3, 5: 1 0 0 2 1 0 0 0 0 0 2 0 0 0 0
%e 4, 6: 1 1 1 0 2 1 0 0 0 2 0 0 0 0 2
%e 5, 7: 1 1 2 0 0 2 1 0 2 0 0 0 0 1 0
%e 6, 8: 1 0 0 1 0 0 2 2 0 0 0 0 0 0 0
%e 7, 10: 1 1 2 0 1 2 0 0 2 1 0 0 2 0 2
%e 8, 11: 1 1 0 0 2 0 2 0 0 2 1 0 0 2 0
%e 9, 12: 1 0 1 1 0 0 0 2 0 0 2 1 2 0 0
%e 10, 13: 1 0 2 2 0 0 0 0 2 0 0 4 1 0 0
%e 11, 14: 1 1 0 0 2 0 1 0 0 2 2 0 2 1 0
%e 12, 15: 1 1 1 0 1 1 2 0 0 1 2 0 0 2 1
%e 13, 17: 1 0 0 0 0 0 0 2 0 0 0 0 2 0 0
%e 14, 18: 1 1 0 0 0 0 2 0 3 0 0 0 0 2 0
%e 15, 19: 1 1 2 0 2 2 0 0 2 2 0 0 0 0 4
%e 16, 20: 1 0 0 1 1 0 0 0 0 0 2 0 0 0 0
%e 17, 21: 1 0 1 2 2 0 1 0 0 0 0 2 0 0 2
%e 18, 22: 1 1 2 0 0 2 2 0 2 0 1 0 2 2 0
%e 19, 23: 1 1 0 0 0 0 2 0 0 0 2 0 2 2 0
%e 20, 24: 1 0 1 1 2 0 0 2 0 0 0 1 0 0 2
%e ...
%e -------------------------------------------------------------
%e The antidiagonals:
%e 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
%e 1: 1
%e 2: 1 1
%e 3: 1 1 0
%e 4: 1 0 1 0
%e 5: 1 1 0 0 0
%e 6: 1 1 1 2 0 0
%e 7: 1 0 2 0 1 1 2
%e 8: 1 1 0 0 2 0 0 0
%e 9: 1 1 2 1 0 1 0 0 0
%e 10: 1 0 0 0 0 2 0 0 0 0
%e 11: 1 0 1 0 1 0 1 0 0 0 0
%e 12: 1 1 2 1 2 2 2 0 0 0 2 0
%e 13: 1 1 0 2 0 0 0 2 2 2 2 0 0
%e 14: 1 0 1 0 0 0 2 0 0 0 0 0 2 2
%e 15: 1 1 0 0 2 0 0 0 2 0 0 0 0 0 0
%e 16: 1 1 0 0 1 0 0 2 0 1 0 0 0 0 0 0
%e 17: 1 0 2 0 0 1 1 0 0 2 0 0 0 0 0 0 2
%e 18: 1 0 0 0 0 0 2 0 2 0 1 0 0 1 2 0 0 0
%e 19: 1 1 1 1 2 0 0 0 0 0 2 0 2 0 0 0 0 0 0
%e 20: 1 1 2 2 1 2 2 2 0 2 0 1 0 0 0 0 0 0 0 0
%e ...
%e For this triangle more of the columns of the array have been used than those that are shown.
%e -----------------------------------------------------------------------------
%e A(2, 3) = 1 because the representative parallel primitive form (rpapf) for discriminant 4*D(2) = 12 and k = +3 is [3, 0, -1], and the one for k= -3 is [-3, 0, 1] (sign flip in both, the a and c entries, but leaving the b entry).
%e A(3, 4) = 2 because the two rpapfs for discriminant 4*D(3) = 20 and k = +4 are [4, 2, -1] and [4, 6, 1], and the two ones for k = -4 are [-4, 2, 1], [-4, 6, -1].
%Y Cf. A000037, A000194, A307236, A307303, A307359, A324252.
%K nonn,tabl
%O 1,19
%A _Wolfdieter Lang_, Apr 21 2019