cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A178628 A (1,1) Somos-4 sequence associated to the elliptic curve E: y^2 - x*y - y = x^3 + x^2 + x.

Original entry on oeis.org

1, 1, -1, -4, -3, 19, 67, -40, -1243, -4299, 25627, 334324, 627929, -29742841, -372632409, 1946165680, 128948361769, 1488182579081, -52394610324649, -2333568937567764, -5642424912729707, 3857844273728205019
Offset: 1

Views

Author

Paul Barry, May 31 2010

Keywords

Comments

a(n) is (-1)^C(n,2) times the Hankel transform of the sequence with g.f.
1/(1-x^2/(1-x^2/(1-4x^2/(1+(3/16)x^2/(1-(76/9)x^2/(1-(201/361)x^2/(1-... where
1,4,-3/16,76/9,201/361,... are the x-coordinates of the multiples of z=(0,0)
on E:y^2-xy-y=x^3+x^2+x.

Links

Crossrefs

Cf. A006720.
(p, q) Somos-4 sequences: A171422, A174168, A174170, A174404, A174809, A174811, A174882, A178075, A178077, A178081, A178079, A178376, A178377, A178384, A178417, A178418, A178621, A178622, A178624, A178625, A178627, A178628, A178644, A184019, A184121, A188313, A188315, A352625.

Programs

  • Magma
    I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2)^2)/Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 18 2018
  • Mathematica
    RecurrenceTable[{a[n] == (a[n-1]*a[n-3] +a[n-2]^2)/a[n-4], a[1] == 1, a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,30}] (* G. C. Greubel, Sep 18 2018 *)
  • PARI
    a(n)=local(E,z);E=ellinit([ -1,1,-1,1,0]);z=ellpointtoz(E,[0,0]); round(ellsigma(E,n*z)/ellsigma(E,z)^(n^2))
  • PARI
    m=30; v=concat([1,1,-1,-4], vector(m-4)); for(n=5, m, v[n] = ( v[n-1]*v[n-3] +v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Sep 18 2018
  • PARI
    {a(n) = subst(elldivpol(ellinit([-1, 1, -1, 1, 0]), n), x ,0)}; /* Michael Somos, Jul 05 2024 */
  • SageMath
    @CachedFunction
def a(n): # a = A178628
    if n<5: return (0,1,1,-1,-4)[n]
    else: return (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4)
[a(n) for n in range(1,41)] # G. C. Greubel, Jul 05 2024

Formula

a(n) = (a(n-1)*a(n-3) + a(n-2)^2)/a(n-4), n>4.
a(n) = -a(-n). a(n) = (-a(n-1)*a(n-4) +4*a(n-2)*a(n-3))/a(n-5) for all n in Z except n=5. - Michael Somos, Jul 05 2024

Extensions

Offset changed to 0. - Michael Somos, Jul 05 2024

A374833 Elliptic net associated to y^2 + y = x^3 + x^2 - 2*x, based on the non-torsion generator points P = [0, 0] and Q = [1, 0].

Original entry on oeis.org

0, 1, 1, -1, 1, -1, -3, 2, 3, -5, -11, -5, 1, 8, 31, 38, 7, -13, -19, 53, 94, -249, 89, 36, -41, 33, 479, -4335, -2357, -149, 181, -151, -350, 919, 5959, 18041, -8767, -4544, 1535, 989, -493, -2591, -12016, 182879, 3085709, 496035, -48259, -11811, -1466, 6627, 13751, -55287, -201383, 5002782, -124991065
Offset: 0

Views

Author

Thomas Scheuerle, Jul 21 2024

Keywords

Comments

The curve y^2 + y = x^3 + x^2 - 2*x is one of the rank-2 elliptic curves with smallest conductor.
The signs are defined by the Weierstrass sigma function. In the literature are other variants of sign assignment for this particular net presented.

Examples

			A(n, k) is a square array read by ascending antidiagonals:
.
      --> k*Q
 n*P |  0,  1,  -1,   -5,   31,    94
  |  |  1,  1,   3,    8,   53,   479
  |  | -1,  2,   1,  -19,   33,   919
 \ / | -3, -5, -13,  -41, -350, -2591
     |-11,  7,  36, -151, -493, 13751
     | 38, 89, 181,  989, 6627, 68428
.
n*P means elliptic point multiplication here. A(n, k)^2 is the denominator of the x coordinate from n*P + k*Q with point multiplication and addition under the elliptic group law for rational numbers.

Links

  • Katherine E. Stange, The Tate Pairing via Elliptic Nets, Proc. of Pairing Conference 2007, T. Takagi et al. eds., LNCS, Vol. 4575, pp. 329-348, Springer-Verlag, Berlin, 2007.

Crossrefs

Cf. A178624.

Programs

  • PARI
    T(n, k) = { local(E, z1); local(E, z2); E=ellinit([0,1,1,-2,0]); z1=ellpointtoz(E,[0,0]); z2=ellpointtoz(E, [1,0]);round(ellsigma(E, n*z1+k*z2)/(ellsigma(E, z1)^(n^2-k*n)*ellsigma(E, z1+z2)^(k*n)*ellsigma(E, z2)^(k^2-k*n) )) }
A(size) = { my(si = max(0, size-5)); M = matconcat([matrix(5,5,m,k,T(m-1,k-1)),matrix(5, si);matrix(si, 5),matrix(si, si)]);
for(k = 1, 5, for(n = 6, size, M[n, k] = (M[n-1, k]*M[n-3, k]*M[3, 1]^2 - M[2, 1]*M[4, 1]*M[n-2, k]^2)/M[n-4, k]));
for(k = 1, 5, for(n = 6, size, M[k, n] = (M[k, n-1]*M[k, n-3]*M[1,3]^2 - M[1, 2]*M[1, 4]*M[k, n-2]^2)/M[k, n-4]));
for(k = 6, size, for(n = 6, size, M[n, k] = (M[n-1, k]*M[n-3, k]*M[3, 1]^2 - M[2, 1]*M[4, 1]*M[n-2, k]^2)/M[n-4, k])) }; M;
sd1(P)=sqrtint(denominator(P[1]));
Pnm(n, m, E, P1, P2) = elladd(E, ellmul(E, P1, n), ellmul(E, P2, m));
Aunsigned(size) = my(E=ellinit([0,1,1,-2,0]), P=[0,0], Q=[1,0]); matrix(size, size, m, n, sd1(Pnm(m-1, n-1, E, P, Q)));

Formula

A(n, k) = ws(z1 + z2)/(ws(z1)^(n^2 - k*n)*ws(z1 + z2)^(k*n)*ws(z2)^(k^2 - k*n)), where ws is the Weierstrass sigma function using the lattice parameters of y^2 + y = x^3 + x^2 - 2*x, z1 is the lattice point corresponding to P = [0, 0] and z2 corresponds to Q = [1, 0].
A(n*c1, n*c2) divides A((n*k)*c1, (n*k)*c2), where c1, c2 are some integer constants not equaling zero simultaneously and k >= 1.
A(n, k) = (A(n-1, k)*A(n-3, k)*A(2, 0)^2 - A(1, 0)*A(3, 0)*A(n-2, k)^2)/A(n-4, k), for n > 4.
A(n, k) = (A(n, k-1)*A(n, k-3)*A(0, 2)^2 - A(0, 1)*A(0, 3)*A(n, k-2)^2)/A(n, k-4), for k > 4.
A(n, n) = (A(n-1, n-1)*A(n-3, n-3)*A(2, 2)^2 - A(1, 1)*A(3, 3)*A(n-2, n-2)^2)/A(n-4, n-4), for n > 4.
|A(n, 0)| = |A178624(n)|.
Showing 1-2 of 2 results.

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