A334068 - OEIS

%I #10 Oct 06 2020 14:42:52

%S 1,0,3,35,-92,8555,162024,2882165,3906507129,-88075171080,

%T 260151768440137,304986999070045520,-100886180199254542253,

%U 1600059682932627475385835,2620000542207768964443625516

%N Negative of numerator of y-coordinate of -(2n+1)*P where P is the generator for rational points on the curve y^2 + y = x^3 - x.

%H Robert Israel, <a href="/A334068/b334068.txt">Table of n, a(n) for n = 0..86</a>

%F a(n) = A028934(-1-n) = A028942(-2*n-1) for all n in Z.

%F 0 = a(n)*a(n+8) -145*a(n+1)*a(n+7) +3225*a(n+2)*a(n+6) -18705*a(n+3)*a(n+5) +14964*a(n+4)*a(n+4) for all n in Z.

%e -P = (0, -1), -3P = (-1 ,0), -5P = (1/4, -3/8), -7P = (-5/9, -35/27).

%p f:= proc(m) option remember; -(-145*procname(m - 7)*procname(m - 1) + 3225*procname(m - 6)*procname(m - 2) - 18705*procname(m - 5)*procname(m - 3) + 14964*procname(m - 4)^2)/procname(m - 8) end proc:

%p Data:= [1, 0, 3, 35, -92, 8555, 162024, 2882165, 3906507129, -88075171080]:

%p for i from 0 to 9 do f(i):= Data[i+1] od:

%p map(f, [$0..20]); # _Robert Israel_, Oct 06 2020

%o (PARI) {a(n) = -numerator(ellmul(ellinit([0, 0, 1, -1, 0]), [0, 0], -2*n-1)[2])};

%Y Cf. A028934, A028942.

%K sign

%O 0,3

%A _Michael Somos_, Apr 13 2020