A028934 - OEIS

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.

%I #28 Apr 13 2020 15:13:06

%S 0,1,5,-8,435,3612,43355,28076979,-331948240,641260644409,

%T 318128427505160,-66316334575107447,588310630753491921045,

%U 435912379274109872312968,2181616293371330311419201915

%N 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.

%H Seiichi Manyama, <a href="/A028934/b028934.txt">Table of n, a(n) for n = 0..86</a>

%H B. Mazur, <a href="https://doi.org/10.1090/S0273-0979-1986-15430-3">Arithmetic on curves</a>, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

%F 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).

%F a(n) = A028942(2n+1). - _Seiichi Manyama_, Nov 20 2016

%F 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

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

%t 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 *)

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

%Y Cf. A028935, A028942.

%K sign,frac

%O 0,3

%A _N. J. A. Sloane_