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 1-3 of 3 results.

A174403 Expansion of (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2).

Original entry on oeis.org

1, 2, 7, 22, 76, 268, 977, 3638, 13804, 53164, 207342, 817212, 3250104, 13026744, 52567461, 213394854, 870845260, 3570590668, 14701822370, 60765209876, 252021314536, 1048538259304, 4375013741962, 18302920281148, 76756814078840, 322618359099896, 1358831330368732
Offset: 0

Views

Author

Paul Barry, Mar 18 2010

Keywords

Comments

G.f. A(x) satisfies A(x)=1+2x*A(x)+2x^2*A(x)+x^2*A(x)^2. Hankel transform is A174404.

Links

Programs

  • Maple
    with(LREtools): with(FormalPowerSeries): # requires Maple 2022
ogf:= (1-2*x-2*x^2-sqrt(1-4*x-4*x^2+8*x^3+4*x^4))/(2*x^2):
req:= FindRE(ogf,x,u(n));
init:= [1, 2, 7, 22, 76, 268]; iseq:= seq(u(i-1)=init[i],i=1..nops(init)):
rmin:= subs(n=n-4,MinimalRecurrence(req,u(n),{iseq})[1]); # Mathar's recurrence
a:= gfun:-rectoproc({rmin, iseq}, u(n), remember):
seq(a(n),n=0..24); # Georg Fischer, Nov 04 2022
with(gfun): # Alternative with gfun alone (use gfun:-version() >= 3.91):
FindSeq := proc(ogf) series(ogf, x, 26): [seq(coeff(%, x, n), n = 0..22)];
listtorec(%, r(n))[1]; subs(n=n-nops(%)-1, %); rectoproc(%, r(n), remember) end:
ogf := (1-sqrt((2*x^2-1)*(2*x*(x+2)-1))-2*x*(x+1))/(2*x^2):
a := FindSeq(ogf): seq(a(n), n=0..28); # Peter Luschny, Nov 04 2022
  • Mathematica
    nmax = 24;
A[_] = 1;
Do[A[x_] = 1 + 2*x*A[x] + 2*x^2*A[x] + x^2*A[x]^2 + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 02 2023 *)

Formula

G.f.: 1/(1-2x-2x^2-x^2/(1-2x-2x^2-x^2/(1-... (continued fraction).
Let A(x) be the g.f., then B(x)=1+x*A(x) = 1 +1*x +2*x^2 +7*x^3 +22*x^4 +... = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-2*x^2) (continued fraction); more generally B(x)=C(x/(1-2*x^2)) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011]
D-finite with recurrence: (n+2)*a(n) -2*(2*n+1)*a(n-1) +4*(1-n)*a(n-2) +4*(2*n-5)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, Sep 30 2012
a(n) ~ 6^(1/4) * (2 + sqrt(6))^(n+1) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 15 2018

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

A179648 Expansion of (1/(1+4x-2x^2))*c(x/(1+4x-2x^2)), c(x) the g.f. of A000108.

Original entry on oeis.org

1, -3, 12, -47, 190, -778, 3224, -13475, 56710, -239986, 1020200, -4353430, 18636908, -80004388, 344264624, -1484499811, 6413133638, -27750688914, 120258432264, -521833284514, 2267084792708, -9859984425324, 42925569027408
Offset: 0

Views

Author

Paul Barry, Jan 09 2011

Keywords

Comments

Hankel transform is the (4,5) Somos-4 sequence A174404.

Links

Crossrefs

Cf. A000108, A174404.

Programs

  • Magma
    m:=50; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1/(2*x))*(1-Sqrt((1-2*x^2)/(1+4*x-2*x^2))))); // G. C. Greubel, Aug 14 2018
  • Mathematica
    CoefficientList[Series[(1/(2*x))*(1 - Sqrt[(1-2*x^2)/(1+4*x-2*x^2)]), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
  • PARI
    x='x+O('x^50); Vec((1/(2*x))*(1-sqrt((1-2*x^2)/(1+4*x-2*x^2)))) \\ G. C. Greubel, Aug 14 2018

Formula

G.f.: (1/(2*x))*(1-sqrt((1-2*x^2)/(1+4*x-2*x^2))) = (sqrt(2*x^2-4*x-1)-sqrt(2*x^2-1))/(2*x*sqrt(2*x^2-4*x-1));
G.f.: 1/(1+4x-2x^2-x/(1-x/(1+4x-2x^2-x/(1-x/(1+4x-2x^2-x/(1-x/(1-... (continued fraction).
Conjecture: (n+1)*a(n) +2*(2n+1)*a(n-1) +4*(1-n)*a(n-2) +4*(5-2n)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 17 2011
a(n) ~ (-1)^n * (2 + sqrt(6))^(n+1) / (2^(3/4) * 3^(1/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 15 2018
Showing 1-3 of 3 results.

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