A307377 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.

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, 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, 2, 1, 2, 2, 2, 0, 0, 0, 2, 0

Offset: 1

Author

Wolfdieter Lang, Apr 21 2019

Keywords

• nonn
• tabl

Comments

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.

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.

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.

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)].

The rpapfs not counted in A324252 and A307303 are equivalent to forms of non-principal cycles for discriminant 4*D(n).

The total number of cycles (the class number h(n)) for discriminant 4*D(n) is given in A307359(n).

The array for the length of the periods of these cycles is given in A307378.

One half of the sum of the length of the periods is given in A307236.

Examples

			The array A(n, k) begins:
n,  D(n) \k  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
-------------------------------------------------------------
1,   2:      1 1 0 0 0 0 2 0 0  0  0  0  0  2  0
2,   3:      1 1 1 0 0 1 0 0 0  0  2  0  2  0  0
3,   5:      1 0 0 2 1 0 0 0 0  0  2  0  0  0  0
4,   6:      1 1 1 0 2 1 0 0 0  2  0  0  0  0  2
5,   7:      1 1 2 0 0 2 1 0 2  0  0  0  0  1  0
6,   8:      1 0 0 1 0 0 2 2 0  0  0  0  0  0  0
7,  10:      1 1 2 0 1 2 0 0 2  1  0  0  2  0  2
8,  11:      1 1 0 0 2 0 2 0 0  2  1  0  0  2  0
9,  12:      1 0 1 1 0 0 0 2 0  0  2  1  2  0  0
10, 13:      1 0 2 2 0 0 0 0 2  0  0  4  1  0  0
11, 14:      1 1 0 0 2 0 1 0 0  2  2  0  2  1  0
12, 15:      1 1 1 0 1 1 2 0 0  1  2  0  0  2  1
13, 17:      1 0 0 0 0 0 0 2 0  0  0  0  2  0  0
14, 18:      1 1 0 0 0 0 2 0 3  0  0  0  0  2  0
15, 19:      1 1 2 0 2 2 0 0 2  2  0  0  0  0  4
16, 20:      1 0 0 1 1 0 0 0 0  0  2  0  0  0  0
17, 21:      1 0 1 2 2 0 1 0 0  0  0  2  0  0  2
18, 22:      1 1 2 0 0 2 2 0 2  0  1  0  2  2  0
19, 23:      1 1 0 0 0 0 2 0 0  0  2  0  2  2  0
20, 24:      1 0 1 1 2 0 0 2 0  0  0  1  0  0  2
...
-------------------------------------------------------------
The antidiagonals:
       1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
1:     1
2:     1 1
3:     1 1 0
4:     1 0 1 0
5:     1 1 0 0 0
6:     1 1 1 2 0 0
7:     1 0 2 0 1 1 2
8:     1 1 0 0 2 0 0 0
9:     1 1 2 1 0 1 0 0 0
10:    1 0 0 0 0 2 0 0 0  0
11:    1 0 1 0 1 0 1 0 0  0  0
12:    1 1 2 1 2 2 2 0 0  0  2  0
13:    1 1 0 2 0 0 0 2 2  2  2  0  0
14:    1 0 1 0 0 0 2 0 0  0  0  0  2  2
15:    1 1 0 0 2 0 0 0 2  0  0  0  0  0  0
16:    1 1 0 0 1 0 0 2 0  1  0  0  0  0  0  0
17:    1 0 2 0 0 1 1 0 0  2  0  0  0  0  0  0  2
18:    1 0 0 0 0 0 2 0 2  0  1  0  0  1  2  0  0  0
19:    1 1 1 1 2 0 0 0 0  0  2  0  2  0  0  0  0  0  0
20:    1 1 2 2 1 2 2 2 0  2  0  1  0  0  0  0  0  0  0  0
...
For this triangle more of the columns of the array have been used than those that are shown.
-----------------------------------------------------------------------------
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).
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].
		

References

• D. A. Buell, Binary Quadratic Forms, Springer, 1989, chapter 3, pp. 21 - 43.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, pp. 112 - 126.

Crossrefs

Cf. A000037, A000194, A307236, A307303, A307359, A324252.