|
| |
|
|
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.
|
|
2
|
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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.
|
|
|
LINKS
|
|
|
|
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.
----------------------------------------------------------------------------
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
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
