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 1-4 of 4 results.

A166914 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 21, a(1) = 340.

Original entry on oeis.org

21, 340, 5456, 87360, 1398016, 22369280, 357912576, 5726617600, 91625947136, 1466015416320, 23456247709696, 375299967549440, 6004799497568256, 96076792028200960, 1537228672719650816, 24595658764588154880
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+2) = 240*a(n).

Links

Crossrefs

Cf. A075153, A166912, A166913, A166915, A166916, A166917, A166927, A166965, A166984.

Programs

  • Magma
    [Binomial(4^(n+3), 2)/96: n in [0..30]]; // G. C. Greubel, Oct 02 2024
  • Mathematica
    CoefficientList[Series[(21-80x)/((1-4x)(1-16x)),{x,0,20}],x]  (* or *) LinearRecurrence[{20,-64},{21,340},20] (* Harvey P. Dale, Feb 23 2011 & Mar 30 2012 *)
  • PARI
    {m=15; v=concat([21, 340], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]); v}
  • SageMath
    A166914=BinaryRecurrenceSequence(20,-64,21,340)
[A166914(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

Formula

a(n) = (64*16^n - 4^n)/3.
G.f.: (21 - 80*x)/((1-4*x)*(1-16*x)).
Limit_{n -> infinity} a(n)/a(n-1) = 16.
From G. C. Greubel, May 28 2016: (Start)
a(n) = 20*a(n-1) - 64*a(n-2).
E.g.f.: (1/3)*(-exp(4*x) + 64*exp(16*x)). (End)

A166917 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 85, a(1) = 1364.

Original entry on oeis.org

85, 1364, 21840, 349504, 5592320, 89478144, 1431654400, 22906486784, 366503854080, 5864061927424, 93824991887360, 1501199874392064, 24019198007050240, 384307168179912704, 6148914691147038720, 98382635059426361344, 1574122160955116748800, 25185954575299047849984
Offset: 0

Views

Author

Klaus Brockhaus, Oct 27 2009

Keywords

Comments

Related to Reverse and Add trajectory of 318 in base 4: A075153(6*n+5) = 240*a(n).

Links

Crossrefs

Cf. A075153, A166912, A166913, A166914, A166915, A166916, A166927, A166965, A166984.

Programs

  • Magma
    [Binomial(4^(n+4), 2)/384: n in [0..30]]; // G. C. Greubel, Oct 02 2024
  • Mathematica
    LinearRecurrence[{20,-64}, {85, 1364}, 50] (* G. C. Greubel, May 28 2016 *)
  • PARI
    {m=15; v=concat([85, 1364], vector(m-2)); for(n=3, m, v[n]=20*v[n-1]-64*v[n-2]); v}
  • SageMath
    A166917=BinaryRecurrenceSequence(20,-64,85,1364)
[A166917(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

Formula

a(n) = (256*16^n - 4^n)/3.
G.f.: (85 - 336*x)/((1-4*x)*(1-16*x)).
Limit_{n -> infinity} a(n)/a(n-1) = 16.
E.g.f.: (1/3)*(256*exp(16*x) - exp(4*x)). - G. C. Greubel, May 28 2016

A166984 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 20.

Original entry on oeis.org

1, 20, 336, 5440, 87296, 1397760, 22368256, 357908480, 5726601216, 91625881600, 1466015154176, 23456246661120, 375299963355136, 6004799480791040, 96076791961092096, 1537228672451215360, 24595658763514413056, 393530540233410478080, 6296488643803287126016
Offset: 0

Views

Author

Klaus Brockhaus, Oct 26 2009

Keywords

Comments

Partial sums of A166965.
First differences of A006105. - Klaus Purath, Oct 15 2020

Links

Crossrefs

Cf. A000302, A002450, A002452, A002453, A006105, A079598, A083584.
Cf. A115490, A166914, A166917, A166927, A166965, A307695.

Programs

  • Magma
    [n le 2 select 19*n-18 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];
  • Mathematica
    LinearRecurrence[{20,-64},{1,20},30] (* Harvey P. Dale, Jul 04 2012 *)
  • PARI
    a(n) = (4*16^n - 4^n)/3 \\ Charles R Greathouse IV, Jun 21 2022
  • SageMath
    A166984=BinaryRecurrenceSequence(20,-64,1,20)
[A166984(n) for n in range(31)] # G. C. Greubel, Oct 02 2024

Formula

a(n) = (4*16^n - 4^n)/3.
G.f.: 1/((1-4*x)*(1-16*x)).
Limit_{n -> oo} a(n)/a(n-1) = 16.
a(n) = A115490(n+1)/3.
Sum_{n>=0} a(n) x^(2*n+4)/(2*n+4)! = ( sinh(x) )^4/4!. - Robert A. Russell, Apr 03 2013
From Klaus Purath, Oct 15 2020: (Start)
a(n) = A002450(n+1)*(A002450(n+2) - A002450(n))/5.
a(n) = (A083584(n+1)^2 - A083584(n)^2)/80. (End)
a(n) = (A079598(n) - A000302(n))/24. - César Aguilera, Jun 21 2022
a(n) = 16*a(n-1) + 4^n with a(0) = 1. - Nadia Lafreniere, Aug 08 2022
E.g.f.: (4/3)*exp(10*x)*sinh(6*x + log(2)). - G. C. Greubel, Oct 02 2024

A166965 a(n) = 20*a(n-1) - 64*a(n-2) for n > 1; a(0) = 1, a(1) = 19.

Original entry on oeis.org

1, 19, 316, 5104, 81856, 1310464, 20970496, 335540224, 5368692736, 85899280384, 1374389272576, 21990231506944, 351843716694016, 5629499517435904, 90071992480301056, 1441151880490123264, 23058430091063197696
Offset: 0

Views

Author

Klaus Brockhaus, Oct 25 2009

Keywords

Comments

lim_{n -> infinity} a(n)/a(n-1) = 16.

Links

Crossrefs

Cf. A166927, A006105 (Gaussian binomial coefficient [ n, 2 ] for q=4).

Programs

  • Magma
    [ n le 2 select 18*n-17 else 20*Self(n-1)-64*Self(n-2): n in [1..17] ];
  • Mathematica
    LinearRecurrence[{20,-64},{1,19},20] (* Harvey P. Dale, Aug 24 2014 *)

Formula

a(n) = (5*16^n - 4^n)/4.
G.f.: (1-x)/((1-4*x)*(1-16*x)).
E.g.f.: (1/4)*(5*exp(16*x) - exp(4*x)). - G. C. Greubel, May 29 2016
Showing 1-4 of 4 results.

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