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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A168049 Expansion of (3 -x -sqrt(1-2*x-3*x^2))/2.

Original entry on oeis.org

1, 0, 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835, 113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019, 142547559, 400763223, 1129760415, 3192727797, 9043402501, 25669818476, 73007772802, 208023278209
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

A variant of the Motzkin numbers A001006. Hankel transform is A168050.
Essentially the same as A086246. - R. J. Mathar, Dec 20 2011
Alternatively, this sequence corresponds to the number of positive walks with n steps {-1,0,1} starting at the origin, ending at altitude 1, and staying strictly above the x-axis. - David Nguyen, Dec 01 2016

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 4*x^5 + 9*x^6 + 21*x^7 + 51*x^8 + ... - _Michael Somos_, Sep 26 2018

Links

Crossrefs

Cf. A001006, A086246, A168050, A168051.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((3 -x - Sqrt(1-2*x-3*x^2))/2)); // G. C. Greubel, Sep 25 2018
  • Mathematica
    CoefficientList[Series[(3-x-Sqrt[1-2*x-3*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)
  • PARI
    Vec((3-x-sqrt(1-2*x-3*x^2))/2) \\ Charles R Greathouse IV, Dec 01 2016

Formula

D-finite with recurrence: n*a(n) +(3-2n)*a(n-1) +3(3-n)*a(n-2)=0. - R. J. Mathar, Dec 20 2011
0 = a(n)*(+9*a(n+1) + 15*a(n+2) - 12*a(n+3)) + a(n+1)*(-3*a(n+1) + 10*a(n+2) - 5*a(n+3)) + a(n+2)*(+a(n+2) + a(n+3)) if n>0. - Michael Somos, Jan 31 2014
a(n) ~ 3^(n+1/2) / (6*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 12 2014
G.f.: 1 + x^2/(1 - x - x^2/(1 - x - x^2/(1 - x - x^2/(1 - ...)))), a continued fraction. - Ilya Gutkovskiy, Sep 23 2017

A172160 a(0)=1. a(n) = 2^(n-2)*(5-n), for n>0.

Original entry on oeis.org

1, 2, 3, 4, 4, 0, -16, -64, -192, -512, -1280, -3072, -7168, -16384, -36864, -81920, -180224, -393216, -851968, -1835008, -3932160, -8388608, -17825792, -37748736, -79691776, -167772160, -352321536, -738197504, -1543503872, -3221225472, -6710886400
Offset: 0

Views

Author

Paul Curtz, Jan 27 2010

Keywords

Comments

The inverse binomial transform is 1,1,0,0,-1,-1,-2,-2,-3,-3 = essentially A168050 or the negative of A004526.

Examples

			G.f. = 1 + 2*x + 3*x^2 + 4*x^3 + 4*x^4 - 16*x^6 - 64*x^7 + ... - _Michael Somos_, Apr 22 2022

Links

Crossrefs

Cf. A004526, A045891, A168050.
Cf. A000079, A001787, A059165, A159964, A131577.

Programs

  • Mathematica
    Table[2^(n-2)*(5-n) -(1/4)*Boole[n==0], {n,0,40}] (* G. C. Greubel, Apr 21 2022 *)
  • SageMath
    [2^(n-2)*(5-n) -(1/4)*bool(n==0) for n in (1..40)] # G. C. Greubel, Apr 21 2022

Formula

a(n+1) - 2*a(n) = -A131577(n).
a(n) + A001787(n-1) = A000079(n+1).
a(n+5) = -A059165(n) = 4*A159964(n+1).
G.f.: (1 - 2*x - x^2)/(1-2*x)^2. - R. J. Mathar, Feb 11 2010
a(n) = 4*a(n-1) - 4*a(n-2), n>2.
E.g.f.: (1/4)*((5-2*x)*exp(2*x) - 1). - G. C. Greubel, Apr 21 2022
a(n) = 4^n*A045891(1-n) if n>1. - Michael Somos, Apr 22 2022

Extensions

Definition replaced with closed form by R. J. Mathar, Feb 11 2010
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