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 1-3 of 3 results.

A168054 Expansion of (1-8x^2-24x^3)/((1-2x)^2*(1+2x+4x^2)).

Original entry on oeis.org

1, 2, -4, -24, -48, -160, -448, -896, -2304, -5632, -11264, -26624, -61440, -122880, -278528, -622592, -1245184, -2752512, -6029312, -12058624, -26214400, -56623104, -113246208, -243269632, -520093696, -1040187392, -2214592512
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

Hankel transform of A168055.

Links

Crossrefs

Cf. A168053, A168055.

Programs

  • Magma
    I:=[1,2,-4,-24]; [n le 4 select I[n] else 2*Self(n-1)+8*Self(n-3)-16*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jul 08 2016
  • Mathematica
    LinearRecurrence[{2, 0, 8, -16}, {1, 2, -4, -24}, 100] (* G. C. Greubel, Jul 07 2016 *)
CoefficientList[Series[(1 - 8 x^2 - 24 x^3) / ((1 - 2 x)^2 (1 + 2 x + 4 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Jul 08 2016 *)

Formula

a(n) = 2^n*A168053(n).
a(n) = 2*a(n-1) + 8*a(n-3) - 16*a(n-4) for n>3. - Vincenzo Librandi, Jul 08 2016

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

A168058 Expansion of x + sqrt(1-2x-3x^2).

Original entry on oeis.org

1, 0, -2, -2, -4, -8, -18, -42, -102, -254, -646, -1670, -4376, -11596, -31022, -83670, -227268, -621144, -1706934, -4713558, -13072764, -36398568, -101704038, -285095118, -801526446, -2259520830, -6385455594, -18086805002
Offset: 0

Views

Author

Paul Barry, Nov 17 2009

Keywords

Comments

a(n+2) = -2*A001006(n). Hankel transform is (-1)^n*A168057(n).
Essentially the same as A167022. - R. J. Mathar, Nov 18 2009

Examples

			1 - 2*x^2 - 2*x^3 - 4*x^4 - 8*x^5 - 18*x^6 - 42*x^7 - 102*x^8 - 254*x^9 - ...

Links

Crossrefs

Cf. A168055.

Programs

  • Mathematica
    CoefficientList[Series[x + Sqrt[1 - 2 x - 3 x^2], {x, 0, 50}], x] (* G. C. Greubel, Jul 08 2016 *)

Formula

a(n) = 0^n - 2*Sum_{k=0..floor((n-2)/2)} C(n-2,2k)*A000108(k).
D-finite with recurrence: n*a(n) +(-2*n+3)*a(n-1) +3*(-n+3)*a(n-2)=0. - R. J. Mathar, Jan 23 2020
Showing 1-3 of 3 results.

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

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