A028942 Negative of numerator of y coordinate of n*P where P is the generator [0,0] for rational points on curve y^2+y = x^3-x.

0, 0, 1, 3, 5, -14, -8, 69, 435, 2065, 3612, -28888, 43355, 2616119, 28076979, -332513754, -331948240, 8280062505, 641260644409, 18784454671297, 318128427505160, -10663732503571536, -66316334575107447, 8938035295591025771

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• frac

Comments

We can take P = P[1] = [x_1, y_1] = [0,0]. Then P[n] = P[1]+P[n-1] = [x_n, y_n] for n >= 2. Sequence gives negated numerators of the y_n. - N. J. A. Sloane, Jan 27 2022

a(n) = A278314(n) up to sign. - Michael Somos, Nov 19 2016

Examples

			3P = (-1, -1),
4P = (2, -3),
5P = (1/4, -5/8),
6P = (6, 14).
		

References

• A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..173
• B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Crossrefs

Cf. A028940, A028941, A028943, A278314.

Programs

PARI

- see A028940.

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).