A028934 Negative of numerator of y-coordinate of (2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

0, 1, 5, -8, 435, 3612, 43355, 28076979, -331948240, 641260644409, 318128427505160, -66316334575107447, 588310630753491921045, 435912379274109872312968, 2181616293371330311419201915

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..86

B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Formula

P = (0, 0), 2P = (1, 0); if kP = (a, b) then (k+1)P = (a' = (b^2 - a^3)/a^2, b' = -1 - b*a'/a).

a(n) = A028942(2n+1). - Seiichi Manyama, Nov 20 2016

0 = a(n)*a(n+8) -145*a(n+1)*a(n+7) +3225*a(n+2)*a(n+6) -18705*a(n+3)*a(n+5) +14964*a(n+4)*a(n+4) for all n in Z. - Michael Somos, Apr 13 2020

Example

3P = (-1, -1). 5P = (1/4, -5/8). 7P = (-5/9, 8/27).

Mathematica

a[ n_] := If[n == 0, 0, -Numerator[ #[[3]]/#[[1]]^3 & @ Nest[Function[z, Module[{w, x, y}, {w, x, y} = z; {w x, y^2 - x^3, -y (y^2 - x^3) - (w x)^3}]], {1, 1, 0}, 2 n - 1]]]; (* Michael Somos, Apr 13 2020 *)

Prog

(PARI) {a(n) = -numerator(ellmul(ellinit([0, 0, 1, -1, 0]), [0, 0], 2*n+1)[2])}; /* Michael Somos, Apr 13 2020 */

Crossrefs

Cf. A028935, A028942.

Sequence in context: A162571 A046490 A155214 * A297559 A195144 A175997

Adjacent sequences: A028931 A028932 A028933 * A028935 A028936 A028937

Keyword

sign,frac

Author

N. J. A. Sloane

Status

approved