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 A307378 #21 Jun 24 2025 13:22:33 %S A307378 2,2,2,2,2,2,4,4,2,2,2,6,2,2,2,2,10,4,4,2,2,2,2,2,2,2,6,6,2,2,6,6,6,6, %T A307378 4,4,2,2,4,4,2,6,2,2,4,4,10,2,2,4,4,8,8,4,4,4,4,4,4,6,6,2,2,2,2,2,6,6, %U A307378 2,2,2,2,6,6,2,2,4,4,6,2,2,2,2,10,10,8,8,6,6,12,12,4,4,2,2,2,2,2,6,2,2,6,6,6,6 %N A307378 Irregular triangle T(n, k) read by rows: row n gives the periods of the cycles of binary quadratic forms of discriminant 4*D(n), with D(n) = A000037(n). %C A307378 The length of row n is 2*A307236(n). This is the number of primitive reduced binary quadratic forms of discriminant 4*D(n), with D(n) = A000037(n). %C A307378 The number of cycles in row n is A307359(n), the class number h(n) of binary quadratic forms of discriminant 4*D(n). %C A307378 The principal cycle starts with F_p(n) = [1, 2*s(n), -(D(n) -s(n))^2], with s(n) = A000194(n). Its period is A307372(n). This is the only cycle (the class number is 1) for n = 1, 3, 10, 13, 24, ... %C A307378 For class number h(n) >= 2 the cycles come mostly in pairs of cycles which can be transformed into each other by a sign flip operation on the outer entries of the forms of the cycle (called outer sign flip). Exceptions occur if cycles are identical with their outer sign flipped ones. This happens, e.g., for n = 7 with two cycles: one of length 2 (the principal cycle CR(2)) and one of length 6. This 6-cycle is also identical to the outer sign flipped one. See the example below. %C A307378 See the Buell and Scholz-Schoeneberg references for cycles and class number, and also the W. Lang link given in A324251, with Table 2. %D A307378 D. A. Buell, Binary Quadratic Forms, Springer, 1989. %D A307378 A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973. %F A307378 T(n, k) = length of k-th cycle of reduced forms of discriminant 4*D(n), with D(n) = A000037(n). %e A307378 The irregular triangle T(n, k) begins: %e A307378 n, D(n) \k 1 2 3 4 ... 2*A307236 %e A307378 --------------------------------------------------- %e A307378 1, 2: 2 2 %e A307378 2, 3: 2 2 4 %e A307378 3, 5: 2 2 %e A307378 4, 6: 2 2 4 %e A307378 5, 7: 4 4 8 %e A307378 6, 8: 2 2 4 %e A307378 7, 10: 2 6 8 %e A307378 8, 11: 2 2 4 %e A307378 9, 12: 2 2 4 %e A307378 10, 13: 10 10 %e A307378 11, 14: 4 4 8 %e A307378 12, 15: 2 2 2 2 8 %e A307378 13, 17: 2 2 %e A307378 14, 18: 2 2 4 %e A307378 15, 19: 6 6 12 %e A307378 16, 20: 2 2 4 %e A307378 17, 21: 6 6 12 %e A307378 18, 22: 6 6 12 %e A307378 19, 23: 4 4 8 %e A307378 20, 24: 2 2 4 4 12 %e A307378 ... %e A307378 --------------------------------------------------- %e A307378 n = 1, D(1) = 2: the only cycle is the principal 2-cycle [[1, 2, -1],[-1, 2, 1]] with discriminant 8. %e A307378 n = 2, D(2) = 3: besides the principal 2-cycle [[1, 2, -2], [-2, 2, 1]] there is another 2-cycle with sign flips in the outer form entries [[2, 2, -1], [-1, 2, 2]], all with discriminant 12. %e A307378 n = 7, D(7) = 10: the principal 2-cycle CR(7) is ([1, 6, -1], [-1, 6, 1]). The other 6-cyle is ([3, 4, -2], [-2, 4, 3], [3, 2, -3], [-3, 4, 2], [2, 4, -3], [-3, 2, 3]). Both cycles are invariant under outer entries sign flips. %Y A307378 Cf. A000037, A000194, A307236, A307359, A307372, A324251. %K A307378 nonn,tabf %O A307378 1,1 %A A307378 _Wolfdieter Lang_, Apr 21 2019