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A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

%I A028936 #23 Mar 23 2022 16:19:21
%S A028936 1,2,6,21,161,1357,18526,480106,12551561,683916417,51678803961,
%T A028936 4881674119706,997454379905326,213822353304561757,
%U A028936 79799551268268089761,53139223644814624290821,36631192030206080565822006,54202648602164057575419038802
%N A028936 Numerator of x-coordinate of (2n)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.
%H A028936 Seiichi Manyama, <a href="/A028936/b028936.txt">Table of n, a(n) for n = 1..106</a>
%H A028936 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 A028936 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 A028936 a(n) = A028940(2n). - _Seiichi Manyama_, Nov 19 2016
%F A028936 0 = a(n)*a(n+6) - 5*a(n+1)*a(n+5) + 4*a(n+2)*a(n+4) - 20*a(n+3)^2 for all n in Z. a(n) = A006720(n+1)*A006720(n+2). - _Michael Somos_, Apr 12 2020
%e A028936 4P =(2, -3).
%e A028936 a(3) = 6 = 2*3 = A006720(4)*A006720(5). - _Michael Somos_, Apr 12 2020
%o A028936 (PARI) a(n)=numerator(ellmul(E,[0,0],2*n)[1]) \\ _Charles R Greathouse IV_, Mar 23 2022
%Y A028936 Cf. A028937 (denominator), A028938, A028939, A028940.
%Y A028936 Cf. A006720.
%K A028936 nonn,frac
%O A028936 1,2
%A A028936 _N. J. A. Sloane_