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.

A102898 A Catalan-related transform of 3^n.

Original entry on oeis.org

1, 3, 9, 30, 99, 330, 1098, 3660, 12195, 40650, 135486, 451620, 1505358, 5017860, 16726068, 55753560, 185844771, 619482570, 2064940470, 6883134900, 22943778138, 76479260460, 254930851404, 849769504680, 2832564956814
Offset: 0

Views

Author

Paul Barry, Jan 17 2005

Keywords

Comments

Transform of 1/(1-3*x) under the mapping g(x) -> g(x*c(x^2)), where c(x) is the g.f. of the Catalan numbers A000108. The inverse transform is h(x) -> h(x/(1+x^2)).

References

  • Maria Paola Bonacina and Nachum Dershowitz, Canonical Inference for Implicational Systems, in Automated Reasoning, Lecture Notes in Computer Science, Volume 5195/2008, Springer-Verlag.

Links

Crossrefs

Cf. A000108, A098615, A100087.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( 2*x/(3*Sqrt(1-4*x^2)+2*x-3) )); // G. C. Greubel, Jul 08 2022
  • Mathematica
    CoefficientList[Series[2*x/(3*Sqrt[1-4*x^2]+2*x-3), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)
  • SageMath
    [1]+[2*sum(k*binomial(n-1, (n-k)//2)*((n-k+1)%2)*3^k/(n+k) for k in (0..n)) for n in (1..40)] # G. C. Greubel, Jul 08 2022

Formula

G.f.: 2*x/(3*sqrt(1-4*x^2) + 2*x - 3).
a(n) = Sum_{k=0..n} k*binomial(n-1, (n-k)/2)*(1 + (-1)^(n-k))*3^k/(n+k), n > 0, with a(0) = 1.
3*n*a(n) - 10*n*a(n-1) - 12*(n-3)*a(n-2) + 40*(n-3)*a(n-3) = 0. - R. J. Mathar, Sep 21 2012
a(n) ~ 2^(n+2) * 5^(n-1) / 3^n. - Vaclav Kotesovec, Feb 01 2014

A100088 Expansion of (1-x^2)/((1-2*x)*(1+x^2)).

Original entry on oeis.org

1, 2, 2, 4, 10, 20, 38, 76, 154, 308, 614, 1228, 2458, 4916, 9830, 19660, 39322, 78644, 157286, 314572, 629146, 1258292, 2516582, 5033164, 10066330, 20132660, 40265318, 80530636, 161061274, 322122548, 644245094, 1288490188, 2576980378
Offset: 0

Views

Author

Paul Barry, Nov 03 2004

Keywords

Comments

A Chebyshev transform of A100087, under the mapping A(x) -> ((1-x^2)/(1+x^2)) * A(x/(1+x^2)).
A176742(n+2) = A084099(n+2) = period 4:repeat 0, -2, 0, 2.

Links

Crossrefs

Cf. A007910, A084099, A100087, A122117, A176742.

Programs

  • Magma
    [n le 3 select Floor((n+2)/2) else 2*Self(n-1) - Self(n-2) +2*Self(n-3): n in [1..41]]; // G. C. Greubel, Jul 08 2022
  • Mathematica
    CoefficientList[Series[(1-x^2)/((1-2x)(1+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,-1,2},{1,2,2},40] (* Harvey P. Dale, May 12 2011 *)
  • Python
    def A100088(n): return ((4<Chai Wah Wu, Apr 22 2025
  • SageMath
    def b(n): return (2/5)*(3*2^(2*n-1) + (-1)^n) # b=A122117
def A100088(n): return b(n/2) if (n%2==0) else 2*b((n-1)/2)
[A100088(n) for n in (0..60)]  # G. C. Greubel, Jul 08 2022

Formula

a(n) = (3*2^n + 2*cos(Pi*n/2) + 4*sin(Pi*n/2))/5.
a(n) = n*Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-1)^k*A100087(n-2*k)/(n-k).
a(n) = 2*a(n-1) + period 4:repeat 0, -2, 0, 2, with a(0) = 1.
a(n) = A007910(n+1) - A007910(n-1).
a(n) = 2*a(n-1) - a(n-2) + 2*a(n-3).
a(n) = (1/5)*(3*2^n + i^n*(1+(-1)^n) - 2*i^(n+1)*(1-(-1)^n)). - G. C. Greubel, Jul 08 2022
a(n) = A122117(n/2) if (n mod 2 = 0) otherwise 2*A122117((n-1)/2). - G. C. Greubel, Jul 21 2022
Showing 1-2 of 2 results.

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