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.

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A162546 A Somos-4 variant: a(n) = (36*a(n-1)*a(n-3) - 68*a(n-2)^2)/a(n-4).

Original entry on oeis.org

1, 1, -16, -644, -40592, -4821056, 17059328, 2492895195136, 10659285907800064, 86296767700623425536, 1081586547380924161458176, -36649408809924048998874742784, -18144416387824430577315746611724288
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform of A162543, A162548.

Links

Crossrefs

Cf. A157005, A157101, A162547.

Programs

  • GAP
    a:=[1,1,-16,-644];; for n in [5..20] do a[n]:=(36*a[n-1]*a[n-3] - 68*a[n-2]^2)/a[n-4]; od; a; # G. C. Greubel, Feb 23 2019
  • Magma
    I:=[1,1,-16,-644]; [n le 4 select I[n] else (36*Self(n-1) *Self(n-3) - 68*Self(n-2)^2)/Self(n-4): n in [1..20]]; // G. C. Greubel, Feb 23 2019
  • Mathematica
    RecurrenceTable[{a[n]==(36*a[n-1]*a[n-3] - 68*a[n-2]^2)/a[n-4], a[0]==1, a[1]==1, a[2]==-16, a[3]==-644}, a, {n,20}] (* G. C. Greubel, Feb 23 2019 *)
  • PARI
    m=20; v=concat([1,1,-16,-644], vector(m-4)); for(n=5, m, v[n] = (36*v[n-1]*v[n-3] -68*v[n-2]^2)/v[n-4]); v \\ G. C. Greubel, Feb 23 2019
  • Sage
    def a(n):
    if (n==0): return 1
    elif (n==1): return 1
    elif (n==2): return -16
    elif (n==3): return -644
    else: return (36*a(n-1)*a(n-3) - 68*a(n-2)^2)/a(n-4)
[a(n) for n in (1..20)] # G. C. Greubel, Feb 23 2019

A162543 A Chebyshev transform of the large Schroeder numbers A006318.

Original entry on oeis.org

1, 2, 5, 18, 73, 312, 1391, 6406, 30235, 145478, 710951, 3519248, 17608681, 88914250, 452512229, 2318774506, 11953427329, 61948592936, 322570037543, 1686777086942, 8854240330363, 46638995523598, 246443050810895
Offset: 0

Views

Author

Paul Barry, Jul 05 2009

Keywords

Comments

Hankel transform is the Somos-4 variant A162546.

Links

Crossrefs

Cf. A162548.

Programs

  • GAP
    a:=[2,5,18,73,312,1391];; for n in [7..30] do a[n]:=(3*(2*n-1)*a[n-1] - (4*n-5)*a[n-2] +12*(n-2)*a[n-3] -(4*n-11)*a[n-4] +3*(2*n-7)*a[n-5] -(n-5)*a[n-6])/(n+1); od; Concatenation([1], a); # G. C. Greubel, Feb 23 2019
  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-x+x^2 - Sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2)) )); // G. C. Greubel, Feb 23 2019
  • Mathematica
    CoefficientList[Series[(1-x+x^2 - Sqrt[1-6*x+3*x^2-6*x^3+x^4])/(2*x*(1+x^2)), {n,0,30}], x] (* G. C. Greubel, Feb 23 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))) \\ G. C. Greubel, Feb 23 2019
  • Sage
    ((1-x+x^2 -sqrt(1-6*x+3*x^2-6*x^3+x^4))/( 2*x*(1+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 23 2019

Formula

G.f.: (1/(1+x^2))*S(x/(1+x^2)), S(x) the g.f. of A006318;
G.f.: (1-x+x^2 - sqrt(1-6*x+3*x^2-6*x^3+x^4))/(2*x*(1+x^2)).
G.f.: 1/(1+x^2-2*x/(1-x/(1+x^2-2*x/(1-x/(1+x^2-2*x/(1-x/(1+x+2*x^2/(1-... (continued fraction);
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n-k,k)*A006318(n-2*k).
Recurrence: (n+1)*a(n) = (5-n)*a(n-6) + 3*(2*n-7)*a(n-5) + (11-4*n)*a(n-4) + 12*(n-2)*a(n-3) + (5-4*n)*a(n-2) + 3*(2*n-1)*a(n-1), n>=6. - Fung Lam, Feb 19 2014
Showing 1-2 of 2 results.

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