A332688 - OEIS

%I #9 Feb 20 2020 08:30:03

%S 0,1,1,-5,-23,181,1740,-8749,-1247497,-33863035,4642247999,

%T 675108007991,-68049162697560,-39309850524879431,4258387436512005121,

%U 10156867585318107377275,4534892371184114325167497,-27717062339512733060005291331,-41962430757987463447351232692380

%N Reduced division polynomials associated with elliptic curve y^2 = x^3 + 17 and multiples of point (-2, 3).

%C For the elliptic curve y^2 = x^3 + 17, the multiples of the point (-2, 3) are (u(n)/a(n)^2, v(n)/a(n)^3) as rationals.

%H LMFDB, <a href="http://www.lmfdb.org/EllipticCurve/Q/10404/f/2">Elliptic Curve 10404.f2 (Cremona label 10404c1)</a>.

%F a(n) = -a(-n), a(n+4)*a(n-3) = -23*a(n+3)*a(n-2) + 348*a(n+1)*a(n) for all n in Z.

%F a(n+12)*a(n-12) = (a(12)/a(6))^2 * a(n+6)*a(n-6) - a(18)/a(6) * a(n)^2 for all n in Z.

%e a(4) = -23 and the point multiple 4*(-2,3) = (752/(-23)^2, 54239/(-23)^3).

%o (PARI) {a(n) = sign(n) * subst(elldivpol(ellinit([0, 17]), abs(n)), x, -2) / (if(n%2, 1, 6) * 2^(n^2\3) * 3^(n^2\4))};

%o (PARI) {a(n) = my(v, s); s=sign(n); n=abs(n); if(n==0, return(0)); v = concat([1, 1, -5, -23, 181, 1740, -8749], vector(max(0, n-7))); for(k=8, n, v[k] = (-23*v[k-1]*v[k-6] +348*v[k-3]*v[k-4])/v[k-7]); s*v[n]};

%o (PARI) {a(n) = my(v, s); s=sign(n); n=abs(n); if(n==0, return(0)); v = concat([1, 1, -5, -23], vector(max(0, n-4))); for(k=5, n, v[k] = (if(k%2, 3, 1)*v[k-1]*v[k-3] +if(k%3==2, 10, 5)*v[k-2]^2)/v[k-4]); s*v[n]};

%o (PARI) {a(n) = my(v, s); s=sign(n); n=abs(n); if(n==0, return(0)); v = concat([1, 1, -5, -23, 181], vector(max(0, n-5))); for(k=6, n, v[k] = (-if(k%3==1, 10, 5)*v[k-1]*v[k-4] +23*v[k-2]*v[k-3])/v[k-5]); s*v[n]};

%K sign

%O 0,4

%A _Michael Somos_, Feb 19 2020