A242107 Reduced division polynomials associated with elliptic curve y^2 + x*y = x^3 - x^2 - x + 1 and multiples of point (0, 1).

0, 1, 1, 1, 1, -1, 2, -3, 1, -5, -7, -13, -16, 11, -57, 131, -113, 389, 670, 2311, 3983, 9, 23647, -81511, 140576, -484247, -833503, -5751815, -14871471, -17124617, -147165662, 710017141, -2273917871, 9644648819, 11396432249, 204006839259, 808162720720

Offset

0,7

Comments

This sequence is similar to Somos-5 (A006721).

For the elliptic curve "58a1" and point (0, 1) the multiple n*(0, 1) = ((3-(-1)^n)/2 * a(n+1)*a(n-1) / a(n)^2, a(n+2)^2 * a(n-4) / a(n)^3). - Michael Somos, Feb 23 2020

Links

Chai Wah Wu, Table of n, a(n) for n = 0..320

LMFDB, Elliptic Curve with LMFDB label 58.a1 (Cremona label 58a1).

Formula

a(n) = -(-1)^n * a(-n) for all n in Z.

0 = a(n)*a(n+5) + a(n+1)*a(n+4) - a(n+2)*a(n+3) for all n in Z.

0 = a(n)*a(n+7) - a(n+1)*a(n+6) - 2*a(n+2)*a(n+5) for all n in Z.

0 = a(n)*a(n+4) + a(n+1)*a(n+3) - a(n+2)*a(n+2) for all even n in Z.

0 = a(n)*a(n+4) + 2*a(n+1)*a(n+3) - a(n+2)*a(n+2) for all odd n in Z.

abs(a(n)) = A242108(n) for all n in Z.

a(2*n) = A178622(n) for all n in Z. - Michael Somos, Aug 21 2014

a(2*n-3) = A328380(n) for all n in Z. - Michael Somos, Feb 23 2020

Example

a(9) = -5 and the point multiple 9*(0, 1) = (-14/(-5)^2, -169/(-5)^3).

Mathematica

Join[{0}, RecurrenceTable[{a[n] == (-a[n-1]*a[n-4] + a[n-2]*a[n-3])/a[n-5], a[0] == 0, a[1] == 1, a[2] == 1, a[3] == 1, a[4] == 1, a[5] == -1}, a, {n, 0, 50}]] (* G. C. Greubel, Aug 05 2018 *)

Prog

(PARI) {a(n) = my(s=1, v); if( n<0, s=-1; n=-n); s^(n+1) * if( n, v = vector(n, k, 1); if( n<6, (-1)^(n>4), v[5] = -1; for(k=6, n, v[k] = (-v[k-1] * v[k-4] + v[k-2] * v[k-3]) / v[k-5]); v[n]))};
(PARI) {a(n) = sign(n) * subst(elldivpol(ellinit([1, -1, 0, -1, 1]), abs(n)), x, 0) / (if(n%2, 1, 2) * (-1)^((n-1)\2) * 2^(n^2\4))}; /* Michael Somos, Feb 23 2020 */
(PARI) {a(n) = my(E=ellinit([1, -1, 0, -1, 1]), z=ellpointtoz(E, [0, 1])); (-1)^(n\2) * round(ellsigma(E, n*z) / (ellsigma(E, z)^n^2 * 2^(n^2\4))) }; /* Michael Somos, Feb 25 2020 */
(Python)
from gmpy2 import divexact
A242107 = [0, 1, 1, 1, 1, -1]
for n in range(6, 321):
....A242107.append(divexact(-A242107[n-1]*A242107[n-4]+A242107[n-2]*A242107[n-3], A242107[n-5])) # Chai Wah Wu, Aug 15 2014
(Magma) I:=[1, 1, 1, 1, -1]; [0] cat [n le 5 select I[n] else (-Self(n-1)* Self(n-4) + Self(n-2)*Self(n-3))/Self(n-5): n in [1..30]]; // G. C. Greubel, Aug 05 2018

Crossrefs

Cf. A006721, A178622, A242108, A328380.

Sequence in context: A358881 A197701 A292770 * A242108 A238941 A247582

Adjacent sequences: A242104 A242105 A242106 * A242108 A242109 A242110

Keyword

sign

Author

Michael Somos, Aug 15 2014

Extensions

Definition edited by Michael Somos, Feb 23 2020

Status

approved