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.

A083102 a(n) = 2*a(n-1) + 10*a(n-2), with a(0) = 1, a(1) = 2.

Original entry on oeis.org

1, 2, 14, 48, 236, 952, 4264, 18048, 78736, 337952, 1463264, 6306048, 27244736, 117549952, 507547264, 2190594048, 9456660736, 40819261952, 176205131264, 760602882048, 3283257076736, 14172542973952, 61177656715264, 264080743170048, 1139938053492736, 4920683538685952
Offset: 0

Views

Author

Mario Catalani (mario.catalani(AT)unito.it), Apr 22 2003

Keywords

Comments

a(n+1) = a(n) + A083101(n). A083101(n)/a(n) converges to sqrt(11).
Antidiagonals of A038207. - Mark Dols, Aug 31 2009
Numerators of stationary probabilities for M2/M/1 queue system. In this queue, customers arrive in groups of 2. Intensity of arrival = 2. Service rate = 5. There is only one server and an infinite queue. - Igor Kleiner, Nov 02 2018

Links

Crossrefs

Cf. A038207, A083101.

Programs

  • Magma
    I:=[1,2]; [n le 2 select I[n] else 2*Self(n-1) + 10*Self(n-2): n in [1..30]]; // G. C. Greubel, Jan 08 2018
  • Mathematica
    CoefficientList[Series[1/(1-2x-10x^2), {x, 0, 25}], x]
LinearRecurrence[{2,10}, {1,2}, 30] (* G. C. Greubel, Jan 08 2018 *)
  • PARI
    x='x+O('x^30); Vec(1/(1-2*x-10*x^2)) \\ G. C. Greubel, Jan 08 2018
  • Sage
    [lucas_number1(n,2,-10) for n in range(1, 24)] # Zerinvary Lajos, Apr 22 2009

Formula

G.f.: 1/(1-2*x-10*x^2).
From Paul Barry, Sep 29 2004: (Start)
E.g.f.: exp(x) * sinh(sqrt(11)*x) / sqrt(11).
a(n) = Sum_{k=0..n} binomial(n,2*k+1) * 11^k. (End)
a(n) = ((1+sqrt(11))^n - (1-sqrt(11))^n)/(2*sqrt(11)). - Rolf Pleisch, Jul 06 2009
G.f.: G(0)/(2-2*x), where G(k)= 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013
G.f.: Q(0)/2 , where Q(k) = 1 + 1/(1 - x*(4*k+2 + 10*x )/( x*(4*k+4 + 10*x ) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 21 2013

A133294 a(n) = 2*a(n-1) + 10*a(n-2), a(0)=1, a(1)=1.

Original entry on oeis.org

1, 1, 12, 34, 188, 716, 3312, 13784, 60688, 259216, 1125312, 4842784, 20938688, 90305216, 389997312, 1683046784, 7266066688, 31362601216, 135385869312, 584397750784, 2522654194688, 10889285897216, 47005113741312
Offset: 0

Views

Author

Philippe Deléham, Dec 20 2007

Keywords

Comments

Binomial transform of [1, 0, 11, 0, 121, 0, 1331, 0, 14641, 0, ...]=: powers of 11 (A001020) with interpolated zeros. - Philippe Deléham, Dec 02 2008
A083101 is an essentially identical sequence (with a different start). - N. J. A. Sloane, Dec 31 2012

Links

Crossrefs

Cf. A083101, A090042.

Programs

  • GAP
    a:=[1,1];; for n in [3..30] do a[n]:=2*a[n-1]+10*a[n-2]; od; a; # G. C. Greubel, Aug 02 2019
  • Magma
    I:=[1,1]; [n le 2 select I[n] else 2*Self(n-1) +10*Self(n-2): n in [1..30]]; // G. C. Greubel, Aug 02 2019
  • Mathematica
    a[n_]:= Simplify[((1+Sqrt[11])^n + (1-Sqrt[11])^n)/2]; Array[a, 30, 0] (* Or *) CoefficientList[Series[(1-x)/(1-2x-10x^2), {x,0,30}], x] (* Or *) LinearRecurrence[{2, 10}, {1, 1}, 30] (* Robert G. Wilson v, Sep 18 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-x)/(1-2*x-10*x^2)) \\ G. C. Greubel, Aug 02 2019
  • Sage
    ((1-x)/(1-2*x-10*x^2)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Aug 02 2019

Formula

a(n) = Sum_{k=0..n} A098158(n,k)*11^(n-k).
G.f.: (1-x)/(1-2*x-10*x^2).
a(n) = A083101(n-1) for n >= 1.
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(11*k-1)/( x*(11*k+10) - 1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 14 2013

Extensions

Terms a(23) onward added by G. C. Greubel, Aug 02 2019
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.