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%I #16 May 07 2019 15:29:48
%S 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,
%T 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,
%U 2,0,0,2,2,0,0,0,2,0
%N 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.
%C 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.
%C The D(n) values for nonzero entries in column k = 1 are given in A003814 (representation of -1).
%C The position list for nonzero entries in row n = 1 is A057126 (conjecture).
%D D. A. Buell, Binary Quadratic Forms, Springer, 1989.
%D A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973.
%F 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.
%e The array A(n, k) begins:
%e n, D(n) \k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e -------------------------------------------------------------------
%e 1, 2: 1 1 0 0 0 0 2 0 0 0 0 0 0 2 0
%e 2, 3: 0 1 1 0 0 0 0 0 0 0 2 0 0 0 0
%e 3, 5: 1 0 0 2 1 0 0 0 0 0 2 0 0 0 0
%e 4, 6: 0 1 0 0 2 1 0 0 0 0 0 0 0 0 2
%e 5, 7: 0 0 2 0 0 2 1 0 0 0 0 0 0 1 0
%e 6, 8: 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0
%e 7, 10: 1 0 0 0 0 2 0 0 2 1 0 0 0 0 2
%e 8, 11: 0 1 0 0 0 0 2 0 0 2 1 0 0 0 0
%e 9, 12: 0 0 1 0 0 0 0 2 0 0 2 1 0 0 0
%e 10, 13: 1 0 2 2 0 0 0 0 2 0 0 4 1 0 0
%e 11, 14: 0 0 0 0 2 0 1 0 0 2 0 0 2 1 0
%e 12, 15: 0 0 0 0 0 1 0 0 0 0 2 0 0 2 1
%e 13, 17: 1 0 0 0 0 0 0 2 0 0 0 0 2 0 0
%e 14, 18: 0 1 0 0 0 0 0 0 2 0 0 0 0 2 0
%e 15, 19: 0 1 2 0 0 0 0 0 0 2 0 0 0 0 4
%e 16, 20: 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0
%e 17, 21: 0 0 1 0 2 0 0 0 0 0 0 2 0 0 0
%e 18, 22: 0 1 0 0 0 0 2 0 0 0 0 0 2 0 0
%e 19, 23: 0 0 0 0 0 0 0 0 0 0 2 0 0 2 0
%e 20, 24: 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2
%e -------------------------------------------------------------------
%e The triangle T(n, k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ..
%e 1: 1
%e 2: 0 1
%e 3: 1 1 0
%e 4: 0 0 1 0
%e 5: 0 1 0 0 0
%e 6: 0 0 0 2 0 0
%e 7: 1 0 2 0 1 0 2
%e 8: 0 0 0 0 2 0 0 0
%e 9: 0 1 0 1 0 1 0 0 0
%e 10: 1 0 0 0 0 2 0 0 0 0
%e 11: 0 0 1 0 0 0 1 0 0 0 0
%e 12: 0 0 2 0 0 2 2 0 0 0 2 0
%e 13: 1 0 0 2 0 0 0 1 0 0 2 0 0
%e 14: 0 0 0 0 0 0 2 0 0 0 0 0 0 2
%e 15: 0 1 0 0 2 0 0 0 2 0 0 0 0 0 0
%e 16: 0 1 0 0 0 0 0 2 0 1 0 0 0 0 0 0
%e 17: 0 0 2 0 0 1 1 0 0 2 0 0 0 0 0 0 2
%e 18: 0 0 0 0 0 0 0 0 2 0 1 0 0 1 2 0 0 0
%e 19: 0 1 1 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0
%e 20: 0 0 0 0 0 0 0 2 0 2 0 1 0 0 0 0 0 0 0 0
%e ...
%e For this triangle more than the shown columns of the array have been used.
%e ----------------------------------------------------------------------------
%e 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.
%Y Cf. A000037, A000194, A003814, A057126, A324252 (positive k), A324251.
%K nonn,tabl
%O 1,19
%A _Wolfdieter Lang_, Apr 20 2019