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.

A083295 a(n) = (4*2^n + (-8)^n)/5.

Original entry on oeis.org

1, 0, 16, -96, 832, -6528, 52480, -419328, 3355648, -26843136, 214749184, -1717985280, 13743898624, -109951156224, 879609315328, -7036874391552, 56294995394560, -450359962632192, 3602879702106112, -28823037614751744, 230584300922208256
Offset: 0

Views

Author

Paul Barry, Apr 24 2003

Keywords

Comments

Binomial transform of A083294.

Crossrefs

Cf. A083296.

Programs

Formula

a(n) = (4*2^n + (-8)^n)/5.
G.f.: (1+6*x)/((1-2*x)*(1+8*x)).
E.g.f.: (4*exp(2*x) + exp(-8*x))/5.

A083297 a(n) = (4*4^n + (-6)^n)/5.

Original entry on oeis.org

1, 2, 20, 8, 464, -736, 12608, -42880, 388352, -1805824, 12932096, -69203968, 448778240, -2558451712, 15887581184, -93178003456, 567657955328, -3371587993600, 20366966915072, -121652045676544, 732111297314816, -4383871690866688, 26338414517288960
Offset: 0

Views

Author

Paul Barry, Apr 24 2003

Keywords

Comments

Binomial transform of A083296.

Crossrefs

Cf. A083222.

Programs

  • GAP
    List([0..25],n->(4*4^n+(-6)^n)/5); # Muniru A Asiru, Oct 31 2018
  • Magma
    [(4*4^n+(-6)^n)/5: n in [0..30]]; // Vincenzo Librandi, Jun 08 2011
    
  • Maple
    seq(coeff(series((1+4*x)/((1-4*x)*(1+6*x)),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Oct 31 2018
  • Mathematica
    CoefficientList[Series[(1 + 4 x)/((1 - 4 x) (1 + 6 x)), {x, 0, 22}], x] (* Michael De Vlieger, Oct 31 2018 *)
    CoefficientList[Series[(4*Exp[4*x] + Exp[-6*x])/5, {x, 0, 50}], x]*Table[k!, {k, 0, 50}] (* Stefano Spezia, Nov 01 2018 *)
    LinearRecurrence[{-2,24}, {1,2}, 30] (* G. C. Greubel, Nov 07 2018 *)
  • PARI
    first(n) = Vec((1+4*x)/((1-4*x)*(1+6*x)) + O(x^n)) \\ Iain Fox, Oct 31 2018
    
  • PARI
    a(n) = (4*4^n + (-6)^n)/5 \\ Iain Fox, Oct 31 2018
    

Formula

a(n) = (4*4^n + (-6)^n)/5.
G.f.: (1+4*x)/((1-4*x)*(1+6*x)).
E.g.f.: (4*exp(4*x) + exp(-6*x))/5.
a(n) = -2*a(n-1) + 24*a(n-2). - Iain Fox, Oct 31 2018

A201865 Expansion of 1/((1-3*x)*(1+7*x)).

Original entry on oeis.org

1, -4, 37, -232, 1705, -11692, 82573, -575824, 4037329, -28241620, 197750389, -1384075576, 9689060473, -67821828988, 474757585885, -3323288752288, 23263064312737, -162841321048996, 1139889634763461, -7979226281082760, 55854587454363721, -390982101720192844
Offset: 0

Views

Author

Bruno Berselli, Dec 07 2011

Keywords

Crossrefs

Programs

  • Magma
    m:=22; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1/((1-3*x)*(1+7*x))));
    
  • Mathematica
    CoefficientList[Series[1/((1-3*x)*(1+7*x)), {x, 0, 22}], x]
  • Maxima
    makelist(coeff(taylor(1/((1-3*x)*(1+7*x)), x, 0, n), x, n), n, 0, 21);
  • PARI
    Vec(1/((1-3*x)*(1+7*x))+O(x^22))
    

Formula

G.f.: 1/((1-3*x)*(1+7*x)).
a(n) = (3^(n+1)+7*(-7)^n)/10.
a(n) = -4*a(n-1)+21*a(n-2) with n>0, a(-1)=0, a(0)=1.
a(n)-a(n-1) = A083300(n)*(-1)^n.
a(n)+5*a(n-1) = A083296(n) with a(-1)=0.
Showing 1-3 of 3 results.