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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.

%I A350622 #18 Jun 27 2022 23:11:37
%S A350622 0,-1,1,2,-3,-2,21,11,-140,209,1740,-3629,-17139,194438,528157,
%T A350622 -8338438,15659721,665838199,-1524968280,-50443970239,791991662680,
%U A350622 8985658531079,-211327567932999,38581695555082,112336114570262877,-20672037869235082,-59711116955990028899,1404304980033091755971,63734020523767895773980,-2251247528715575677121711,-33058398375463796474831580
%N 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.
%C A350622 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.
%D A350622 D. Husemoller, Elliptic Curves, Springer, 1987, p. 28.
%D A350622 A. W. Knapp, Elliptic Curves, Princeton, 1992, p. 64.
%e A350622 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]
%o A350622 (PARI) \\ To get the first 40 points P[n].
%o A350622 \\ define curve E
%o A350622 E = ellinit([0,1,1,0,0])
%o A350622 P[1] = [0,0]
%o A350622 for(n=2, 40, P[n] = elladd(E, P[1], P[n-1]))
%o A350622 P
%Y A350622 Cf. A028940-A028943, A350623-A350625.
%K A350622 sign,frac
%O A350622 1,4
%A A350622 _N. J. A. Sloane_, Jan 27 2022