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.

A110307 Expansion of (1+2*x)/((1+x+x^2)*(1+5*x+x^2)).

Original entry on oeis.org

1, -4, 17, -80, 384, -1842, 8827, -42292, 202631, -970862, 4651680, -22287540, 106786021, -511642564, 2451426797, -11745491420, 56276030304, -269634660102, 1291897270207, -6189851690932, 29657361184451, -142096954231322, 680827409972160, -3262040095629480
Offset: 0

Views

Author

Creighton Dement, Jul 19 2005

Keywords

Links

Crossrefs

Cf. A002320, A004253, A049347, A099837, A110308, A110309, A110310, A110311.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+2*x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023
  • Maple
    seriestolist(series((1+2*x)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));
  • Mathematica
    LinearRecurrence[{-6,-7,-6,-1}, {1,-4,17,-80}, 41] (* G. C. Greubel, Jan 03 2023 *)
  • PARI
    Vec((1+2*x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019
  • SageMath
    def U(n,x): return chebyshev_U(n, x)
def A110307(n): return (1/4)*(3*U(n,-5/2) +U(n-1,-5/2) +U(n,-1/2) -U(n-1,-1/2))
[A110307(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

Formula

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+1).
a(n) + a(n+1) + a(n+2) = A002320(n).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/4)*(3*U(n,-5/2) + U(n-1,-5/2) + U(n,-1/2) - U(n-1,-1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023

A110308 Expansion of -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)).

Original entry on oeis.org

0, -2, 11, -52, 247, -1182, 5664, -27140, 130037, -623044, 2985181, -14302860, 68529120, -328342742, 1573184591, -7537580212, 36114716467, -173036002122, 829065294144, -3972290468600, 19032387048857, -91189644775684, 436915836829561, -2093389539372120
Offset: 0

Views

Author

Creighton Dement, Jul 19 2005

Keywords

Links

Crossrefs

Cf. A004253, A049347, A099837, A110307, A110309, A110310, A110311.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( -x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023
  • Maple
    seriestolist(series(-x*(2+x)/((x^2+x+1)*(x^2+5*x+1)), x=0,25));
  • Mathematica
    LinearRecurrence[{-6,-7,-6,-1}, {0,-2,11,-52}, 40] (* G. C. Greubel, Jan 03 2023 *)
  • PARI
    concat(0, Vec(-x*(2+x)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25))) \\ Colin Barker, Apr 30 2019
  • SageMath
    def U(n, x): return chebyshev_U(n,x)
def A110308(n): return (1/4)*(2*U(n, -5/2) +U(n-1, -5/2) -2*U(n, -1/2) -U(n-1, -1/2))
[A110308(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

Formula

a(n+2) = - 5*a(n+1) - a(n) - A099837(n+2).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/4)*(2*U(n, -5/2) + U(n-1, -5/2) - 2*U(n, -1/2) - U(n-1, -1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 03 2023

A110309 Expansion of (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)).

Original entry on oeis.org

1, -3, 12, -57, 275, -1320, 6325, -30303, 145188, -695637, 3332999, -15969360, 76513801, -366599643, 1756484412, -8415822417, 40322627675, -193197315960, 925663952125, -4435122444663, 21249948271188, -101814618911277, 487823146285199, -2337301112514720
Offset: 0

Views

Author

Creighton Dement, Jul 19 2005

Keywords

Links

Crossrefs

Cf. A004253, A049347, A109265, A110307, A110308, A110310, A110311.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 03 2023
  • Maple
    seriestolist(series((1+3*x+x^2)/((x^2+5*x+1)*(x^2+x+1)), x=0,25));
  • Mathematica
    LinearRecurrence[{-6,-7,-6,-1}, {1,-3,12,-57}, 40] (* G. C. Greubel, Jan 03 2023 *)
  • PARI
    Vec((1+3*x+x^2)/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, Apr 30 2019
  • SageMath
    def A110309(n): return (1/2)*(chebyshev_U(n,-5/2)+chebyshev_U(n,-1/2))
[A110309(n) for n in range(41)] # G. C. Greubel, Jan 03 2023

Formula

a(n+2) = - 5*a(n+1) - a(n) + (-1)^n*A109265(n+3).
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, Apr 30 2019
a(n) = (1/2)*(ChebyshevU(n, -5/2) + ChebyshevU(n, -1/2)). - G. C. Greubel, Jan 03 2023

A110311 Expansion of 1/((1+x+x^2)*(1+5*x+x^2)).

Original entry on oeis.org

1, -6, 29, -138, 660, -3162, 15151, -72594, 347819, -1666500, 7984680, -38256900, 183299821, -878242206, 4207911209, -20161313838, 96598657980, -462831976062, 2217561222331, -10624974135594, 50907309455639, -243911573142600, 1168650556257360, -5599341208144200
Offset: 0

Views

Author

Creighton Dement, Jul 19 2005

Keywords

Comments

In reference to the program code, A004254(n+1) = 1ibaseiseq[A*B](n).
Superseeker finds: a(n) + a(n+1) + a(n+2) = (-1)^n*A004254(n+3).

Links

Crossrefs

Cf. A004253, A004254, A110307, A110308, A110309, A110310.

Programs

  • Magma
    R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( 1/((1+x+x^2)*(1+5*x+x^2)) )); // G. C. Greubel, Jan 02 2023
  • Maple
    seriestolist(series(1/((x^2+5*x+1)*(x^2+x+1)), x=0,25));
  • Mathematica
    LinearRecurrence[{-6,-7,-6,-1}, {1,-6,29,-138}, 40] (* G. C. Greubel, Jan 02 2023 *)
  • PARI
    Vec(1/((1+x+x^2)*(1+5*x+x^2)) + O(x^25)) \\ Colin Barker, May 14 2019
  • SageMath
    def U(n,x): return chebyshev_U(n,x)
def A110311(n): return (1/4)*(5*U(n, -5/2) + U(n-1, -5/2) - U(n, -1/2) - U(n-1, -1/2))
[A110311(n) for n in range(41)] # G. C. Greubel, Jan 02 2023

Formula

a(n+2) = - 5*a(n+1) - a(n) + ((-1)^n)*A109265(n+1)/2.
a(n) = -6*a(n-1) - 7*a(n-2) - 6*a(n-3) - a(n-4) for n>3. - Colin Barker, May 14 2019
a(n) = (1/4)*(5*U(n, -5/2) + U(n-1, -5/2) - U(n, -1/2) - U(n-1, -1/2)), where U(n, x) = ChebyshevU(n, x). - G. C. Greubel, Jan 02 2023
Showing 1-4 of 4 results.

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