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 A307377 #19 May 17 2022 02:05:10 %S A307377 1,1,1,1,1,0,1,0,1,0,1,1,0,0,0,1,1,1,2,0,0,1,0,2,0,1,1,2,1,1,0,0,2,0, %T A307377 0,0,1,1,2,1,0,1,0,0,0,1,0,0,0,0,2,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,1,1, %U A307377 2,1,2,2,2,0,0,0,2,0 %N A307377 Array A(n, k) read by upwards antidiagonals giving the number of representative parallel primitive binary quadratic forms for discriminant Disc(n) = 4*D(n), with D(n) = A000037(n), and for representable integer |k| >= 1. %C A307377 For the definition of representative parallel primitive forms (rpapfs) for discriminant Disc > 0 (the indefinite 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 A307377 Note that the number of rpapfs of a discriminant Disc > 0 for k >= 1 is identical with the one for negative k. These forms differ in the signs of the a and c entries of these forms but not the b >= 0 entry (called an outer sign flip). See some examples below, and the program in the mentioned W. Lang link, section 3. %C A307377 For the forms counted in the array A(n, k) see Table 3 of the W. Lang link given in A324251, for n = 1..30 and k = 1..10. %C A307377 Compare the present array with the ones given in A324252 and A307303 for the number of rpapfs for discriminant 4*D(n) and representable positive and negative k, respectively, that are equivalent (under SL(2, Z)) to the reduced principal form F_p = [1, 2*s(n), -(D(n) - s(n)^2)] with s(n) = A000194(n), of the unreduced Pell form F(n) = [1, 0, -D(n)]. %C A307377 The rpapfs not counted in A324252 and A307303 are equivalent to forms of non-principal cycles for discriminant 4*D(n). %C A307377 The total number of cycles (the class number h(n)) for discriminant 4*D(n) is given in A307359(n). %C A307377 The array for the length of the periods of these cycles is given in A307378. %C A307377 One half of the sum of the length of the periods is given in A307236. %D A307377 D. A. Buell, Binary Quadratic Forms, Springer, 1989, chapter 3, pp. 21 - 43. %D A307377 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, pp. 112 - 126. %e A307377 The array A(n, k) begins: %e A307377 n, D(n) \k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ... %e A307377 ------------------------------------------------------------- %e A307377 1, 2: 1 1 0 0 0 0 2 0 0 0 0 0 0 2 0 %e A307377 2, 3: 1 1 1 0 0 1 0 0 0 0 2 0 2 0 0 %e A307377 3, 5: 1 0 0 2 1 0 0 0 0 0 2 0 0 0 0 %e A307377 4, 6: 1 1 1 0 2 1 0 0 0 2 0 0 0 0 2 %e A307377 5, 7: 1 1 2 0 0 2 1 0 2 0 0 0 0 1 0 %e A307377 6, 8: 1 0 0 1 0 0 2 2 0 0 0 0 0 0 0 %e A307377 7, 10: 1 1 2 0 1 2 0 0 2 1 0 0 2 0 2 %e A307377 8, 11: 1 1 0 0 2 0 2 0 0 2 1 0 0 2 0 %e A307377 9, 12: 1 0 1 1 0 0 0 2 0 0 2 1 2 0 0 %e A307377 10, 13: 1 0 2 2 0 0 0 0 2 0 0 4 1 0 0 %e A307377 11, 14: 1 1 0 0 2 0 1 0 0 2 2 0 2 1 0 %e A307377 12, 15: 1 1 1 0 1 1 2 0 0 1 2 0 0 2 1 %e A307377 13, 17: 1 0 0 0 0 0 0 2 0 0 0 0 2 0 0 %e A307377 14, 18: 1 1 0 0 0 0 2 0 3 0 0 0 0 2 0 %e A307377 15, 19: 1 1 2 0 2 2 0 0 2 2 0 0 0 0 4 %e A307377 16, 20: 1 0 0 1 1 0 0 0 0 0 2 0 0 0 0 %e A307377 17, 21: 1 0 1 2 2 0 1 0 0 0 0 2 0 0 2 %e A307377 18, 22: 1 1 2 0 0 2 2 0 2 0 1 0 2 2 0 %e A307377 19, 23: 1 1 0 0 0 0 2 0 0 0 2 0 2 2 0 %e A307377 20, 24: 1 0 1 1 2 0 0 2 0 0 0 1 0 0 2 %e A307377 ... %e A307377 ------------------------------------------------------------- %e A307377 The antidiagonals: %e A307377 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ... %e A307377 1: 1 %e A307377 2: 1 1 %e A307377 3: 1 1 0 %e A307377 4: 1 0 1 0 %e A307377 5: 1 1 0 0 0 %e A307377 6: 1 1 1 2 0 0 %e A307377 7: 1 0 2 0 1 1 2 %e A307377 8: 1 1 0 0 2 0 0 0 %e A307377 9: 1 1 2 1 0 1 0 0 0 %e A307377 10: 1 0 0 0 0 2 0 0 0 0 %e A307377 11: 1 0 1 0 1 0 1 0 0 0 0 %e A307377 12: 1 1 2 1 2 2 2 0 0 0 2 0 %e A307377 13: 1 1 0 2 0 0 0 2 2 2 2 0 0 %e A307377 14: 1 0 1 0 0 0 2 0 0 0 0 0 2 2 %e A307377 15: 1 1 0 0 2 0 0 0 2 0 0 0 0 0 0 %e A307377 16: 1 1 0 0 1 0 0 2 0 1 0 0 0 0 0 0 %e A307377 17: 1 0 2 0 0 1 1 0 0 2 0 0 0 0 0 0 2 %e A307377 18: 1 0 0 0 0 0 2 0 2 0 1 0 0 1 2 0 0 0 %e A307377 19: 1 1 1 1 2 0 0 0 0 0 2 0 2 0 0 0 0 0 0 %e A307377 20: 1 1 2 2 1 2 2 2 0 2 0 1 0 0 0 0 0 0 0 0 %e A307377 ... %e A307377 For this triangle more of the columns of the array have been used than those that are shown. %e A307377 ----------------------------------------------------------------------------- %e A307377 A(2, 3) = 1 because the representative parallel primitive form (rpapf) for discriminant 4*D(2) = 12 and k = +3 is [3, 0, -1], and the one for k= -3 is [-3, 0, 1] (sign flip in both, the a and c entries, but leaving the b entry). %e A307377 A(3, 4) = 2 because the two rpapfs for discriminant 4*D(3) = 20 and k = +4 are [4, 2, -1] and [4, 6, 1], and the two ones for k = -4 are [-4, 2, 1], [-4, 6, -1]. %Y A307377 Cf. A000037, A000194, A307236, A307303, A307359, A324252. %K A307377 nonn,tabl %O A307377 1,19 %A A307377 _Wolfdieter Lang_, Apr 21 2019