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A242107 Reduced division polynomials associated with elliptic curve y^2 + x*y = x^3 - x^2 - x + 1 and multiples of point (0, 1). 4

%I #31 Sep 08 2022 08:46:08

%S 0,1,1,1,1,-1,2,-3,1,-5,-7,-13,-16,11,-57,131,-113,389,670,2311,3983,

%T 9,23647,-81511,140576,-484247,-833503,-5751815,-14871471,-17124617,

%U -147165662,710017141,-2273917871,9644648819,11396432249,204006839259,808162720720

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

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

%C 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

%H Chai Wah Wu, <a href="/A242107/b242107.txt">Table of n, a(n) for n = 0..320</a>

%H LMFDB, <a href="https://www.lmfdb.org/EllipticCurve/Q/58/a/1">Elliptic Curve with LMFDB label 58.a1 (Cremona label 58a1)</a>.

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

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

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

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

%F 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.

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

%F a(2*n) = A178622(n) for all n in Z. - _Michael Somos_, Aug 21 2014

%F a(2*n-3) = A328380(n) for all n in Z. - _Michael Somos_, Feb 23 2020

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

%t 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 *)

%o (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]))};

%o (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 */

%o (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 */

%o (Python)

%o from gmpy2 import divexact

%o A242107 = [0,1,1,1,1,-1]

%o for n in range(6,321):

%o ....A242107.append(divexact(-A242107[n-1]*A242107[n-4]+A242107[n-2]*A242107[n-3],A242107[n-5])) # _Chai Wah Wu_, Aug 15 2014

%o (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

%Y Cf. A006721, A178622, A242108, A328380.

%K sign

%O 0,7

%A _Michael Somos_, Aug 15 2014

%E Definition edited by _Michael Somos_, Feb 23 2020

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)