This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A324252 #18 May 08 2019 03:08:43 %S A324252 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 A324252 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 A324252 2,0,0,2,2,0,0,0,2,0,1,0,0,2,0,0,0,1,0,0,2,0,0 %N A324252 Triangle T(n, k) read by rows from 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 A324252 The array A(n, k) gives the number of the representative parallel binary quadratic primitive forms for discriminant Disc(n) = 4*D(n) = 4*A000037(n) and representation of positive integer k which are (properly) equivalent to the Pell form F(n) = [1, 0, -D(n)]. %C A324252 For the definition of representative parallel primitive forms for discriminant Disc > 0 (the indefinte case) and representation of nonzero integers k see the Scholz-Schoeneberg reference, p. 105, or the Buell reference p. 49 (without use of the name parallel). For the procedure to find the primitive representative parallel forms (rpapfs) for Disc(n) = 4*D(n) = 4*A000037(n) and nonzero integer k see the W. Lang link given in A324251, section 3. %C A324252 Among them the parallel forms which are equivalent to the reduced principal form F_p(n) = [1, 2*s(n), -(D(n) - s(n))^2], with s(n) = A000194(n), are important to find all solutions (x, y) with gcd(x, y) = 1 (proper) of the Pell form F(n) = [1, 0, -D(n)] with Disc(F(n)) = 4*D(n) > 0 representing a positive integer k. The number of these parallel forms pa(n, k) gives the number of the proper fundamental solutions. The general solution is obtained from the fundamental solutions with the help of integer powers of the automorphic matrix corresponding to the cycle determined by the reduced principal form F_p(n). %C A324252 Thus the array A(n,k) gives the number of proper families (also called classes) of solutions of the Pell equation x^2 - Dn(n)*y^2 = k, for positive integer k. The positions of the nonzero entries in row n give the list of the k values for which proper solutions exist. %C A324252 These position lists are A057126 (conjecture) and A243655, for k = 1 and 2. %C A324252 The first column has only 1s, showing that every Pell form [1, 0, -D(n)] represents k = +1, and that there is only one family of proper solutions. %D A324252 D. A. Buell, Binary Quadratic Forms, Springer, 1989, chapter 3, pp. 21-43. %D A324252 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, pp. 112-126. %F A324252 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 >= 1, with D(n) = A000037(n), for n >= 1. %e A324252 The array A(n, k) begins: %e A324252 n, D(n) \k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... %e A324252 ------------------------------------------------------------ %e A324252 1, 2: 1 1 0 0 0 0 2 0 0 0 0 0 0 2 0 %e A324252 2, 3: 1 0 0 0 0 1 0 0 0 0 0 0 2 0 0 %e A324252 3, 5: 1 0 0 2 1 0 0 0 0 0 2 0 0 0 0 %e A324252 4, 6: 1 0 1 0 0 0 0 0 0 2 0 0 0 0 0 %e A324252 5, 7: 1 1 0 0 0 0 0 0 2 0 0 0 0 0 0 %e A324252 6, 8: 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 %e A324252 7, 10: 1 0 0 0 0 2 0 0 2 1 0 0 0 0 2 %e A324252 8, 11: 1 0 0 0 2 0 0 0 0 0 0 0 0 2 0 %e A324252 9, 12: 1 0 0 1 0 0 0 0 0 0 0 0 2 0 0 %e A324252 10, 13: 1 0 2 2 0 0 0 0 2 0 0 4 1 0 0 %e A324252 11, 14: 1 1 0 0 0 0 0 0 0 0 2 0 0 0 0 %e A324252 12, 15: 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 %e A324252 13, 17: 1 0 0 0 0 0 0 2 0 0 0 0 2 0 0 %e A324252 14, 18: 1 0 0 0 0 0 2 0 1 0 0 0 0 0 0 %e A324252 15, 19: 1 0 0 0 2 2 0 0 2 0 0 0 0 0 0 %e A324252 16, 20: 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 %e A324252 17, 21: 1 0 0 2 0 0 1 0 0 0 0 0 0 0 2 %e A324252 18, 22: 1 0 2 0 0 0 0 0 2 0 1 0 0 2 0 %e A324252 19, 23: 1 1 0 0 0 0 0 0 0 0 0 0 2 0 0 %e A324252 20, 24: 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 %e A324252 ... %e A324252 ------------------------------------------------------------- %e A324252 The triangle T(n, k) begins: %e A324252 n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... %e A324252 1: 1 %e A324252 2: 1 1 %e A324252 3: 1 0 0 %e A324252 4: 1 0 0 0 %e A324252 5: 1 0 0 0 0 %e A324252 6: 1 1 1 2 0 0 %e A324252 7: 1 0 0 0 1 1 2 %e A324252 8: 1 0 0 0 0 0 0 0 %e A324252 9: 1 0 0 0 0 0 0 0 0 %e A324252 10: 1 0 0 0 0 0 0 0 0 0 %e A324252 11: 1 0 0 0 0 0 0 0 0 0 0 %e A324252 12: 1 1 2 1 2 2 0 0 0 0 0 0 %e A324252 13: 1 0 0 2 0 0 0 1 2 2 2 0 0 %e A324252 14: 1 0 0 0 0 0 0 0 0 0 0 0 2 2 %e A324252 15: 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 %e A324252 16: 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 %e A324252 17: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 %e A324252 18: 1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 %e A324252 19: 1 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 %e A324252 20: 1 1 2 2 1 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 %e A324252 ... For this triangle more of the columns of the array have been used than those that are shown. %e A324252 ---------------------------------------------------------------------------- %e A324252 A(5, 9) = 2 = T(13, 9) because D(5) = 7, and the Pell form F(5) with disc(F(5)) = 4*7 = 28 representing k = +9 has 2 families (classes) of proper solutions generated from the two positive fundamental positive solutions (x10, y10) = (11, 4) and (x20, y20) = (4, 1). They are obtained from the trivial solutions of the parallel forms [9, 8, 1] and [9, 10, 2], respectively. See the W. Lang link in A324251, section 3. %Y A324252 Cf. A000037, A000194, A057126, A243655, A307303 (negative k), A307377, A324251. %K A324252 nonn,tabl %O A324252 1,19 %A A324252 _Wolfdieter Lang_, Apr 20 2019