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.

A083299 a(n) = (4*6^n + (-4)^n)/5.

Original entry on oeis.org

1, 4, 32, 160, 1088, 6016, 38144, 220672, 1356800, 8009728, 48582656, 289398784, 1744781312, 10435133440, 62745018368, 375933239296, 2257746919424, 13537891581952, 81261709230080, 487432816427008, 2925146654375936
Offset: 0

Views

Author

Paul Barry, Apr 24 2003

Keywords

Comments

Binomial transform of A083222.

Crossrefs

Cf. A083300.

Programs

Formula

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

A083301 a(n) = (4*8^n + (-2)^n)/5.

Original entry on oeis.org

1, 6, 52, 408, 3280, 26208, 209728, 1677696, 13421824, 107374080, 858993664, 6871947264, 54975582208, 439804649472, 3518437212160, 28147497664512, 225179981381632, 1801439850921984, 14411518807638016, 115292150460579840
Offset: 0

Views

Author

Paul Barry, Apr 24 2003

Keywords

Comments

Binomial transform of A083300.

Crossrefs

Cf. A083302.

Programs

  • Magma
    [(4*8^n+(-2)^n)/5: n in [0..25]]; // Vincenzo Librandi, Jun 29 2011
    
  • Mathematica
    Table[(4 * 8^n + (-2)^n)/5, {n, 0, 19}] (* Alonso del Arte, Mar 29 2011 *)
  • PARI
    a(n)=4^(n+1)\/5<Charles R Greathouse IV, Jun 29 2011
  • Sage
    [lucas_number1(n,6,-16) for n in range(1, 21)] # Zerinvary Lajos, Apr 24 2009
    

Formula

G.f.: 1/((1-8*x)(1+2*x)).
E.g.f.: (4*exp(8*x) + exp(-2*x))/5.

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.