A350622 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^2.

0, -1, 1, 2, -3, -2, 21, 11, -140, 209, 1740, -3629, -17139, 194438, 528157, -8338438, 15659721, 665838199, -1524968280, -50443970239, 791991662680, 8985658531079, -211327567932999, 38581695555082, 112336114570262877, -20672037869235082, -59711116955990028899, 1404304980033091755971, 63734020523767895773980, -2251247528715575677121711, -33058398375463796474831580

Offset

1,4

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.

References

D. Husemoller, Elliptic Curves, Springer, 1987, p. 28.

A. W. Knapp, Elliptic Curves, Princeton, 1992, p. 64.

Links

Table of n, a(n) for n=1..31.

Example

P[1] to P[16] are [0, 0], [-1, -1], [1, -2], [2, 3], [-3/4, 1/8], [-2/9, -28/27], [21, -99], [11/49, 20/343], [-140/121, -931/1331], [209/400, -10527/8000], [1740/361, 76400/6859], [-3629/7569, 71117/658503], [-17139/36481, -7705242/6967871], [194438/38809, -97805561/7645373], [528157/1036324, 317884519/1054977832], [-8338438/7187761, -6053168484/19270387241]

Prog

(PARI) \\ To get the first 40 points P[n].
\\ define curve E
E = ellinit([0, 1, 1, 0, 0])
P[1] = [0, 0]
for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))
P

Crossrefs

Cf. A028940-A028943, A350623-A350625.

Sequence in context: A101033 A136454 A025522 * A019228 A361531 A332734

Adjacent sequences: A350619 A350620 A350621 * A350623 A350624 A350625

Keyword

sign,frac

Author

N. J. A. Sloane, Jan 27 2022

Status

approved