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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.
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%I #42 Jun 27 2022 01:52:17

%S 0,1,-1,2,1,6,-5,21,-20,161,116,1357,-3741,18526,8385,480106,-239785,

%T 12551561,-59997896,683916417,1849037896,51678803961,-270896443865,

%U 4881674119706,-16683000076735,997454379905326

%N 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.

%C 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

%D A. W. Knapp, Elliptic Curves, Princeton 1992, p. 77.

%H Seiichi Manyama, <a href="/A028940/b028940.txt">Table of n, a(n) for n = 1..212</a>

%H B. Mazur, <a href="https://doi.org/10.1090/S0273-0979-1986-15430-3">Arithmetic on curves</a>, Bull. Amer. Math. Soc. 14 (1986), 207-259; see p. 225.

%F 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).

%F a(-n) = a(n) = - A006769(n-1) * A006769(n+1) for all n in Z. - _Michael Somos_, Jul 28 2016

%e 4P = P[4] = [2, -3].

%e 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

%o (PARI) \\ from _N. J. A. Sloane_, Jan 27 2022. To get the first 40 points P[n].

%o \\ define curve E

%o E = ellinit([0,0,1,-1,0]) \\ y^2+y = x^3-x

%o P = vector(100)

%o P[1] = [0,0]

%o for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))

%o P

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

%K sign,frac

%O 1,4

%A _N. J. A. Sloane_