OFFSET
1,3
COMMENTS
This gives the row lengths of the irregular triangle A343232.
This sequence gives the number of representative parallel primitive forms (rpapfs) of the positive definite binary quadratic form F = x^2 + x*y + y^2 (with Discriminant Disc = -3) representing positive integers k. Only certain odd k, namely k = k(n) = A034017(n+1), for n >= 1, have proper solutions F = k.
Positive definite binary quadratic primitive forms F = [a, b, c], with a > 0 and gcd(a, b, c) = 1, with odd discriminants Disc = b^2 - 4*a*c = -D < 0, that is, D == 3 (mod 4), and representation of positive integers k have representative parallel primitive forms (rpapfs) Fpa(D,k;j) = [k, 2*j+1, (j^2 + j + (D+1)/4)/k].
Each rpapf produces a trivial proper solution to F = k, obtained from the trivial solution of Fpa(D,k;j) = k by (x, y) = (1,0), via equivalence transformations of determinant +1 achieved by applying the inverse of products of matrices R(t) = Mat([0,-1], [1t]]) for certain values t. The R(t) transformations are used to obtain from a primitive form F = [a, b, c] the equivalent so-called unique half-reduced (right) neighbor form F' = [c, -b + 2*c*t, a - b*t + c*t^2], with the choice t = ceiling((b/c - 1)/2). (c > 0 because a > 0 for positive definite forms with D > 0.)
FORMULA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 08 2021
STATUS
approved