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 A307303 #16 May 07 2019 15:29:48 %S A307303 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 A307303 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 A307303 2,0,0,2,2,0,0,0,2,0 %N 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. %C A307303 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 A307303 The D(n) values for nonzero entries in column k = 1 are given in A003814 (representation of -1). %C A307303 The position list for nonzero entries in row n = 1 is A057126 (conjecture). %D A307303 D. A. Buell, Binary Quadratic Forms, Springer, 1989. %D A307303 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973. %F A307303 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 A307303 The array A(n, k) begins: %e A307303 n, D(n) \k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... %e A307303 ------------------------------------------------------------------- %e A307303 1, 2: 1 1 0 0 0 0 2 0 0 0 0 0 0 2 0 %e A307303 2, 3: 0 1 1 0 0 0 0 0 0 0 2 0 0 0 0 %e A307303 3, 5: 1 0 0 2 1 0 0 0 0 0 2 0 0 0 0 %e A307303 4, 6: 0 1 0 0 2 1 0 0 0 0 0 0 0 0 2 %e A307303 5, 7: 0 0 2 0 0 2 1 0 0 0 0 0 0 1 0 %e A307303 6, 8: 0 0 0 1 0 0 2 1 0 0 0 0 0 0 0 %e A307303 7, 10: 1 0 0 0 0 2 0 0 2 1 0 0 0 0 2 %e A307303 8, 11: 0 1 0 0 0 0 2 0 0 2 1 0 0 0 0 %e A307303 9, 12: 0 0 1 0 0 0 0 2 0 0 2 1 0 0 0 %e A307303 10, 13: 1 0 2 2 0 0 0 0 2 0 0 4 1 0 0 %e A307303 11, 14: 0 0 0 0 2 0 1 0 0 2 0 0 2 1 0 %e A307303 12, 15: 0 0 0 0 0 1 0 0 0 0 2 0 0 2 1 %e A307303 13, 17: 1 0 0 0 0 0 0 2 0 0 0 0 2 0 0 %e A307303 14, 18: 0 1 0 0 0 0 0 0 2 0 0 0 0 2 0 %e A307303 15, 19: 0 1 2 0 0 0 0 0 0 2 0 0 0 0 4 %e A307303 16, 20: 0 0 0 1 0 0 0 0 0 0 2 0 0 0 0 %e A307303 17, 21: 0 0 1 0 2 0 0 0 0 0 0 2 0 0 0 %e A307303 18, 22: 0 1 0 0 0 0 2 0 0 0 0 0 2 0 0 %e A307303 19, 23: 0 0 0 0 0 0 0 0 0 0 2 0 0 2 0 %e A307303 20, 24: 0 0 0 0 0 0 0 1 0 0 0 0 0 0 2 %e A307303 ------------------------------------------------------------------- %e A307303 The triangle T(n, k) begins: %e A307303 n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 .. %e A307303 1: 1 %e A307303 2: 0 1 %e A307303 3: 1 1 0 %e A307303 4: 0 0 1 0 %e A307303 5: 0 1 0 0 0 %e A307303 6: 0 0 0 2 0 0 %e A307303 7: 1 0 2 0 1 0 2 %e A307303 8: 0 0 0 0 2 0 0 0 %e A307303 9: 0 1 0 1 0 1 0 0 0 %e A307303 10: 1 0 0 0 0 2 0 0 0 0 %e A307303 11: 0 0 1 0 0 0 1 0 0 0 0 %e A307303 12: 0 0 2 0 0 2 2 0 0 0 2 0 %e A307303 13: 1 0 0 2 0 0 0 1 0 0 2 0 0 %e A307303 14: 0 0 0 0 0 0 2 0 0 0 0 0 0 2 %e A307303 15: 0 1 0 0 2 0 0 0 2 0 0 0 0 0 0 %e A307303 16: 0 1 0 0 0 0 0 2 0 1 0 0 0 0 0 0 %e A307303 17: 0 0 2 0 0 1 1 0 0 2 0 0 0 0 0 0 2 %e A307303 18: 0 0 0 0 0 0 0 0 2 0 1 0 0 1 2 0 0 0 %e A307303 19: 0 1 1 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 %e A307303 20: 0 0 0 0 0 0 0 2 0 2 0 1 0 0 0 0 0 0 0 0 %e A307303 ... %e A307303 For this triangle more than the shown columns of the array have been used. %e A307303 ---------------------------------------------------------------------------- %e A307303 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 A307303 Cf. A000037, A000194, A003814, A057126, A324252 (positive k), A324251. %K A307303 nonn,tabl %O A307303 1,19 %A A307303 _Wolfdieter Lang_, Apr 20 2019