A307303 Triangle T(n, k) read as upwards antidiagonals of array A, where A(n, k) is the number of families (also called classes) of proper solutions of the Pell equation x^2 - D(n)*y^2 = -k, for k >= 1.

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

Offset: 1

Author

Wolfdieter Lang, Apr 20 2019

Keywords

• nonn
• tabl

Comments

For details see A324252 which gives the array for the numbers of families of proper solutions of x^2 - D(n)*y^2 = k for positive integers k. See also the W. Lang link in A324251, section 3.

The D(n) values for nonzero entries in column k = 1 are given in A003814 (representation of -1).

The position list for nonzero entries in row n = 1 is A057126 (conjecture).

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:      0  1  1  0  0  0  0  0  0  0  2  0  0  0  0
3,   5:      1  0  0  2  1  0  0  0  0  0  2  0  0  0  0
4,   6:      0  1  0  0  2  1  0  0  0  0  0  0  0  0  2
5,   7:      0  0  2  0  0  2  1  0  0  0  0  0  0  1  0
6,   8:      0  0  0  1  0  0  2  1  0  0  0  0  0  0  0
7,  10:      1  0  0  0  0  2  0  0  2  1  0  0  0  0  2
8,  11:      0  1  0  0  0  0  2  0  0  2  1  0  0  0  0
9,  12:      0  0  1  0  0  0  0  2  0  0  2  1  0  0  0
10, 13:      1  0  2  2  0  0  0  0  2  0  0  4  1  0  0
11, 14:      0  0  0  0  2  0  1  0  0  2  0  0  2  1  0
12, 15:      0  0  0  0  0  1  0  0  0  0  2  0  0  2  1
13, 17:      1  0  0  0  0  0  0  2  0  0  0  0  2  0  0
14, 18:      0  1  0  0  0  0  0  0  2  0  0  0  0  2  0
15, 19:      0  1  2  0  0  0  0  0  0  2  0  0  0  0  4
16, 20:      0  0  0  1  0  0  0  0  0  0  2  0  0  0  0
17, 21:      0  0  1  0  2  0  0  0  0  0  0  2  0  0  0
18, 22:      0  1  0  0  0  0  2  0  0  0  0  0  2  0  0
19, 23:      0  0  0  0  0  0  0  0  0  0  2  0  0  2  0
20, 24:      0  0  0  0  0  0  0  1  0  0  0  0  0  0  2
-------------------------------------------------------------------
The triangle T(n, k) begins:
n\k   1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ..
1:    1
2:    0  1
3:    1  1  0
4:    0  0  1  0
5:    0  1  0  0  0
6:    0  0  0  2  0  0
7:    1  0  2  0  1  0  2
8:    0  0  0  0  2  0  0  0
9:    0  1  0  1  0  1  0  0  0
10:   1  0  0  0  0  2  0  0  0  0
11:   0  0  1  0  0  0  1  0  0  0  0
12:   0  0  2  0  0  2  2  0  0  0  2  0
13:   1  0  0  2  0  0  0  1  0  0  2  0  0
14:   0  0  0  0  0  0  2  0  0  0  0  0  0  2
15:   0  1  0  0  2  0  0  0  2  0  0  0  0  0  0
16:   0  1  0  0  0  0  0  2  0  1  0  0  0  0  0  0
17:   0  0  2  0  0  1  1  0  0  2  0  0  0  0  0  0  2
18:   0  0  0  0  0  0  0  0  2  0  1  0  0  1  2  0  0  0
19:   0  1  1  1  0  0  0  0  0  0  2  0  0  0  0  0  0  0  0
20:   0  0  0  0  0  0  0  2  0  2  0  1  0  0  0  0  0  0  0  0
...
For this triangle more than the shown columns of the array have been used.
----------------------------------------------------------------------------
A(5, 6) = 2 = T(10, 6)  because D(5) =  7, and the Pell form F(5) with disc(F(5)) = 4*7 = 28 representing k = -6 has 2 families (classes) of proper solutions generated from the two positive fundamental positive solutions (x10, y10) = (13, 5) and  (x20, y20) = (1, 1). They are obtained from the trivial solutions of the parallel forms [-6, 2, 1] and [-6, 10, -3], respectively.
		

References

• D. A. Buell, Binary Quadratic Forms, Springer, 1989.
• A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973.

Crossrefs

Cf. A000037, A000194, A003814, A057126, A324252 (positive k), A324251.

Formula

T(n, k) = A(n-k+1, k) for 1 <= k <= n, with A(n,k) the number of proper (positive) fundamental solutions of the Pell equation x^2 - D(n)*y^2 = -k for k >= 1, with D(n) = A000037(n), for n >= 1. Each such fundamental solution generates a family of proper solutions.