A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

1, 2, 6, 21, 161, 1357, 18526, 480106, 12551561, 683916417, 51678803961, 4881674119706, 997454379905326, 213822353304561757, 79799551268268089761, 53139223644814624290821, 36631192030206080565822006, 54202648602164057575419038802

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..106

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) = A028940(2n). - Seiichi Manyama, Nov 19 2016

0 = a(n)*a(n+6) - 5*a(n+1)*a(n+5) + 4*a(n+2)*a(n+4) - 20*a(n+3)^2 for all n in Z. a(n) = A006720(n+1)*A006720(n+2). - Michael Somos, Apr 12 2020

Example

4P =(2, -3).
a(3) = 6 = 2*3 = A006720(4)*A006720(5). - Michael Somos, Apr 12 2020

Prog

(PARI) a(n)=numerator(ellmul(E, [0, 0], 2*n)[1]) \\ Charles R Greathouse IV, Mar 23 2022

Crossrefs

Cf. A028937 (denominator), A028938, A028939, A028940.

Cf. A006720.

Sequence in context: A356015 A110306 A351691 * A066932 A181754 A367678

Adjacent sequences: A028933 A028934 A028935 * A028937 A028938 A028939

Keyword

nonn,frac

Author

N. J. A. Sloane

Status

approved