OFFSET
1,19
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).
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.
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.
EXAMPLE
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.
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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.
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Wolfdieter Lang, Apr 20 2019
STATUS
approved