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.

A249997 Expansion of 1/((1-x)*(1+3*x)*(1-4*x)).

Original entry on oeis.org

1, 2, 15, 40, 221, 702, 3355, 11780, 52041, 193402, 817895, 3138720, 12953461, 50618102, 206059635, 813476860, 3286192481, 13047914802, 52482224575, 209057202200, 838843897101, 3347530323502, 13413657088715, 53584020970740, 214547906035321, 857556157684202
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016208, A099621, A249998, A249999.

Programs

  • Magma
    [((-1)^n*3^(n+3) +4^(n+3) -7)/84: n in [0..50]]; // G. C. Greubel, Jul 21 2022
  • Mathematica
    LinearRecurrence[{2,11,-12}, {1,2,15}, 50] (* G. C. Greubel, Jul 21 2022 *)
  • SageMath
    [((-1)^n*3^(n+3) +4^(n+3) -7)/84 for n in (0..50)] # G. C. Greubel, Jul 21 2022

Formula

G.f.: 1/((1-x) * (1+3*x) * (1-4*x)).
a(n) = (-1)^n*3^(n+2)/28 + 4^(n+2)/21 -1/12. - R. J. Mathar, Jan 09 2015
E.g.f.: (1/84)*(27*exp(-3*x) - 7*exp(x) + 64*exp(4*x)). - G. C. Greubel, Jul 21 2022

A298564 a(n) = (3^(n+2)+11)/2 - 5*2^(n+1) + 2*n.

Original entry on oeis.org

0, 1, 10, 53, 218, 789, 2658, 8581, 26986, 83477, 255506, 776709, 2350554, 7092565, 21359554, 64242437, 193054922, 579820053, 1740770802, 5224933765, 15680044090, 47050617941, 141172825250, 423560418693, 1270765142058, 3812463198229, 11437725138898, 34313846505221, 102942881692826
Offset: 0

Views

Author

M. F. Hasler, Jan 21 2018

Keywords

Comments

Partial sums of A281773; first differences of A285361.

Links

Crossrefs

Cf. A281773, A285361.

Programs

  • Mathematica
    Array[(3^(# + 2) + 11)/2 - 5*2^(# + 1) + 2 # &, 29, 0] (* or *)
CoefficientList[Series[x (1 + 3 x)/((3 x - 1) (2 x - 1) (x - 1)^2), {x, 0, 28}], x] (* Michael De Vlieger, Jan 21 2018 *)
  • PARI
    A298564(n)=2*n-5<<(n+1)+3^(n+2)\2+5
  • Python
    def A298564list(n):
    def generator():
        a, b, c = 5, 3, 0
        while True:
            yield c
            a *= 2
            b *= 3
            c += 2 - a + b
    a = generator()
    return [next(a) for _ in range(n)]
print(A298564list(29)) # Peter Luschny, Jan 22 2018

Formula

G.f.: x*(1+3*x) / ( (3*x-1)*(2*x-1)*(x-1)^2 ). - R. J. Mathar, Jan 21 2018
a(n) = A249999(n-1) +3*A249999(n-2). - R. J. Mathar, Jan 21 2018
Showing 1-2 of 2 results.

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