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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, Einf├╝hrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973.

LINKS

Table of n, a(n) for n=1..78.

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

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

Sequence in context: A239003 A123759 A072453 * A324252 A321445 A007423

Adjacent sequences:  A307300 A307301 A307302 * A307304 A307305 A307306

KEYWORD

nonn,tabl

AUTHOR

Wolfdieter Lang, Apr 20 2019

STATUS

approved

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Last modified December 10 23:29 EST 2019. Contains 329910 sequences. (Running on oeis4.)