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.

A168073 Expansion of 1 + 3*(1-x-sqrt(1-2*x-3*x^2))/2.

Original entry on oeis.org

1, 0, 3, 3, 6, 12, 27, 63, 153, 381, 969, 2505, 6564, 17394, 46533, 125505, 340902, 931716, 2560401, 7070337, 19609146, 54597852, 152556057, 427642677, 1202289669, 3389281245, 9578183391, 27130207503, 77009455428, 219023318406, 624069834627, 1781228354487
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Comments

Hankel transform is A168072. a(n+2)=3*A000106(n). Another variant is A168076.

Links

Crossrefs

Cf. A168055, A168049.
Cf. A000106, A168072, A168076.

Programs

  • Mathematica
    CoefficientList[Series[1 + 3*(1 - x - Sqrt[1 - 2*x - 3*x^2])/2, {x, 0, 50}], x] (* G. C. Greubel, Jul 09 2016 *)

Formula

a(n) = 0^n+3*Sum_{k=0..floor((n-2)/2)} C(n-2,2k)*A000108(k).
D-finite with recurrence: a(n) = ((2*n-3)*a(n-1)+(3*n-9)*a(n-2))/n for n>=3, a(0)=1, a(1)=0, a(2)=3. - Sergei N. Gladkovskii, Jul 16 2012

A168075 Expansion of (1+27x^2-54x^3)/((1+3x)^2*(1-3x+9 x^2)).

Original entry on oeis.org

1, -3, 36, -189, 567, -2430, 9477, -28431, 104976, -373977, 1121931, -3897234, 13286025, -39858075, 133923132, -444816117, 1334448351, -4390765542, 14334558093, -43003674279, 139471376040, -449795187729, 1349385563187, -4330586226042, 13839047287569
Offset: 0

Views

Author

Paul Barry, Nov 18 2009

Keywords

Comments

Hankel transform of A168076.

Links

Crossrefs

Cf. A061891, A168076.

Programs

  • Magma
    I:=[1,-3,36,-189]; [n le 4 select I[n] else -3*Self(n-1)-27*Self(n-3)-81*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jul 10 2016
  • Mathematica
    LinearRecurrence[{-3, 0, -27, -81}, {1, -3, 36, -189}, 50] (* G. C. Greubel, Jul 09 2016 *)
CoefficientList[Series[(1 + 27 x^2 - 54 x^3) / ((1 + 3 x)^2 (1 - 3 x + 9 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 10 2016 *)
  • PARI
    Vec((1+27*x^2-54*x^3)/((1+3*x)^2*(1-3*x+9*x^2))+ O(x^30)) \\ Michel Marcus, Dec 03 2014

Formula

a(n) = (-3)^n*A061891(n).
a(n) = 2*(-3)^n*n + 3^n*(sin(Pi*n/3)/sqrt(3) + cos(Pi*n/3)). - Ilya Gutkovskiy, Jul 10 2016

Extensions

Corrected by R. J. Mathar, Dec 03 2014
Showing 1-2 of 2 results.

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

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