%I A127863
%S A127863 1,2,5,0,2,4,8,0,5,10,7,0,1,0,13,0,18,4,0,0,1,8,5,0,7,16,4,0,0,0,10,0,
%T A127863 14,10,13,0,17,20,0,0,11,14,19,0,40,0,7,0,0,2,19,0,11,0,17,0,9,26,25,0,
%U A127863 19,0,0,0,23,36,28,0,0,8,16,0,35,0,5,0,29,0,0,0,31,2,16,0,0,16,5,0,0,10
%V A127863 1,-2,5,0,2,4,8,0,-5,-10,-7,0,-1,0,-13,0,18,-4,0,0,-1,-8,5,0,-7,-16,-4,
               0,0,0,10,0,14,
%W A127863 10,-13,0,17,20,0,0,-11,14,-19,0,40,0,-7,0,0,2,-19,0,11,0,17,0,-9,26,-25,
               0,-19,0,0,0,
%X A127863 23,-36,-28,0,0,8,-16,0,-35,0,5,0,29,0,0,0,-31,2,16,0,0,16,-5,0,0,-10
%N A127863 L-series coefficients for elliptic curve E243b1: y^2+y=x^3+2.
%F A127863 a(n)=b(3n+1) where b(n) is multiplicative and b(3^e) = 0^e, b(p^e) = 
               (1+(-1)^e)/2*(-p)^(e/2) if p == 2 (mod 3), b(p^e) = b(p)*b(p^(e-1)) 
               -p*b(p^(e-2)) if p == 1 (mod 3) where b(p) = -sum(x=0..p-1, kronecker(4*x^3+9, 
               p)).
%F A127863 a(4n+3)=0.
%e A127863 q - 2*q^4 + 5*q^7 + 2*q^13 + 4*q^16 + 8*q^19 - 5*q^25 - 10*q^28 - ...
%o A127863 (PARI) {a(n)=if(n<0, 0, ellak( ellinit( [0,0,1,0,2]), 3*n+1))}
%o A127863 (PARI) {a(n)=local(A, p, e, x, y, a0, a1); if(n<0, 0, n=3*n+1; A=factor(n); 
               prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3, 0, a0=1; 
               a1=y=-sum(x=0, p-1, kronecker(4*x^3+9, p)); for(i=2, e, x=y*a1-p*a0; 
               a0=a1; a1=x); a1)))) }
%Y A127863 Sequence in context: A118349 A011183 A005671 this_sequence A006891 A054675 
               A136209
%Y A127863 Adjacent sequences: A127860 A127861 A127862 this_sequence A127864 A127865 
               A127866
%K A127863 sign
%O A127863 0,2
%A A127863 Michael Somos, Feb 03 2007