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

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

Offset

1,4

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

G. C. Greubel, Table of n, a(n) for n = 1..160

Paul Barry, Riordan arrays, the A-matrix, and Somos 4 sequences, arXiv:1912.01126 [math.CO], 2019.

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

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 *)

Prog

(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 */
(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
(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

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.

Sequence in context: A183231 A241358 A178417 * A345125 A167479 A278424

Adjacent sequences: A178625 A178626 A178627 * A178629 A178630 A178631

Keyword

easy,sign

Author

Paul Barry, May 31 2010

Extensions

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

Status

approved