A028940 a(n) = numerator of the X-coordinate of n*P where P is the generator [0,0] for rational points on the curve y^2 + y = x^3 - x.

0, 1, -1, 2, 1, 6, -5, 21, -20, 161, 116, 1357, -3741, 18526, 8385, 480106, -239785, 12551561, -59997896, 683916417, 1849037896, 51678803961, -270896443865, 4881674119706, -16683000076735, 997454379905326

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 numerators of the x_n. - N. J. A. Sloane, Jan 27 2022

Examples

			4P = P[4] = [2, -3].
P[1] to P[16] are [0, 0], [1, 0], [-1, -1], [2, -3], [1/4, -5/8], [6, 14], [-5/9, 8/27], [21/25, -69/125], [-20/49, -435/343], [161/16, -2065/64], [116/529, -3612/12167], [1357/841, 28888/24389], [-3741/3481, -43355/205379], [18526/16641, -2616119/2146689], [8385/98596, -28076979/30959144], [480106/4225, 332513754/274625]. - _N. J. A. Sloane_, Jan 27 2022
		

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..212
• B. Mazur, Arithmetic on curves, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

Crossrefs

Cf. A006769, A028941, A028942, A028943, A350622-A350625.

Programs

PARI

\\ from N. J. A. Sloane, Jan 27 2022. To get the first 40 points P[n].
\\ define curve E
E = ellinit([0,0,1,-1,0]) \\ y^2+y = x^3-x
P = vector(100)
P[1] = [0,0]
for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))
P

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) = a(n) = - A006769(n-1) * A006769(n+1) for all n in Z. - Michael Somos, Jul 28 2016