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A278314 a(n) = -c(n-1) * c(n-2) * c(n+3) where c(n) = A006769(n).

%I A278314 #6 Nov 19 2016 05:32:48
%S A278314 0,0,1,-3,-5,-14,-8,69,-435,2065,3612,28888,-43355,-2616119,28076979,
%T A278314 -332513754,331948240,8280062505,641260644409,18784454671297,
%U A278314 -318128427505160,10663732503571536,-66316334575107447,-8938035295591025771,-588310630753491921045
%N A278314 a(n) = -c(n-1) * c(n-2) * c(n+3) where c(n) = A006769(n).
%C A278314 a(n) = A028942(n) up to sign.
%C A278314 y coordinate of n*P = -A028942(n) / A028943(n) = a(n) / A006769(n)^3 where P is generator for rational points on curve y^2 + y = x^3 - x.
%H A278314 Seiichi Manyama, <a href="/A278314/b278314.txt">Table of n, a(n) for n = 1..173</a>
%F A278314 0 = a(n)*a(n+8) - a(n+1)*a(n+7) - 3*a(n+2)*a(n+6) + 3*a(n+3)*a(n+5) - 6*a(n+4)^2 for all n in Z.
%F A278314 0 = a(n+1)*a(n+2)*a(n+6) - 2*a(n+1)*a(n+3)*a(n+5) + 3*a(n+1)*a(n+4)^2 + 3*a(n+2)^2*a(n+5) + a(n+2)*a(n+3)*a(n+4) - a(n+3)^3 for all n in Z.
%e A278314 G.f. = x^3 - 3*x^4 - 5*x^5 - 14*x^6 - 8*x^7 + 69*x^8 - 435*x^9 + ...
%o A278314 (PARI) {a(n) = my(m, an); if( n>0, m = n; an = vector( max(12, m), i, if( i<13, [0, 0, 1, -3, -5, -14, -8, 69, -435, 2065, 3612, 28888][i], 0)), m = 1-n; an = vector( max(12, m), i, if( i<13, [1, 1, 1, 0, -2, 3, -15, -35, -56, -92, 2001, -8555][i], 0))); for( k=13, m, an[k] = (an[k-1] * an[k-7] + 3 *  an[k-2] * an[k-6] - 3 * an[k-3] * an[k-5] + 6 * an[k-4]^2) / an[k-8]); an[m]};
%Y A278314 Cf. A006769, A028942, A028943.
%K A278314 sign
%O A278314 1,4
%A A278314 _Michael Somos_, Nov 17 2016