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Search: id:A127863
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| A127863 |
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L-series coefficients for elliptic curve E243b1: y^2+y=x^3+2. |
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+0
1
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| 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, 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, 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
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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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)).
a(4n+3)=0.
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EXAMPLE
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q - 2*q^4 + 5*q^7 + 2*q^13 + 4*q^16 + 8*q^19 - 5*q^25 - 10*q^28 - ...
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PROGRAM
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(PARI) {a(n)=if(n<0, 0, ellak( ellinit( [0, 0, 1, 0, 2]), 3*n+1))}
(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)))) }
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CROSSREFS
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Sequence in context: A118349 A011183 A005671 this_sequence A006891 A054675 A136209
Adjacent sequences: A127860 A127861 A127862 this_sequence A127864 A127865 A127866
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KEYWORD
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sign
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AUTHOR
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Michael Somos, Feb 03 2007
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