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%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_
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