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
KEYWORD
sign,frac
AUTHOR
STATUS
approved