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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, 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 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.)