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.

A248225 a(n) = 6^n - 3^n.

Original entry on oeis.org

0, 3, 27, 189, 1215, 7533, 45927, 277749, 1673055, 10058013, 60407127, 362619909, 2176250895, 13059099693, 78359381127, 470170635669, 2821066860735, 16926530304573, 101559569247927, 609358577749029, 3656154953278575, 21936940180024653
Offset: 0

Views

Author

Vincenzo Librandi, Oct 04 2014

Keywords

Links

Crossrefs

Cf. sequences of the form k^n-3^n: A005061 (k=4), A005058 (k=5), this sequence (k=6), A190541 (k=7), A190543 (k=8), A059410 (k=9), A248226 (k=10), A139741 (k=11).
Cf. A000225, A000244.

Programs

  • Magma
    [6^n-3^n: n in [0..30]];
  • Mathematica
    Table[6^n - 3^n, {n, 0, 25}] (* or *) CoefficientList[Series[3 x / ((1 - 3 x) (1 - 6 x)), {x, 0, 30}], x]
LinearRecurrence[{9,-18},{0,3},30] (* Harvey P. Dale, Jul 12 2025 *)

Formula

G.f.: 3*x/((1-3*x)*(1-6*x)).
a(n) = 9*a(n-1) - 18*a(n-2).
a(n) = 3^n*(2^n - 1) = A000244(n)*A000225(n).
E.g.f.: 2*exp(9*x/2)*sinh(3*x/2). - Elmo R. Oliveira, Mar 31 2025

A016146 Expansion of g.f. 1/((1-3*x)*(1-11*x)).

Original entry on oeis.org

1, 14, 163, 1820, 20101, 221354, 2435623, 26794040, 294741001, 3242170694, 35663936683, 392303480660, 4315338818701, 47468728600034, 522156019383343, 5743716227565680, 63180878546269201, 694989664138101374, 7644886305906535603, 84093749366134153100, 925031243030962468501
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A139741.

Programs

  • Mathematica
    Join[{a=1,b=14},Table[c=14*b-33*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
CoefficientList[Series[1/((1-3x)(1-11x)),{x,0,30}],x] (* or *) LinearRecurrence[ {14,-33},{1,14},30] (* Harvey P. Dale, Dec 18 2018 *)

Formula

a(n) = 14*a(n-1) - 33*a(n-2), n >= 2. - Vincenzo Librandi, Mar 14 2011
a(n) = (-3^(n+1) + 11^(n+1))/8. - R. J. Mathar, Mar 15 2011
From Elmo R. Oliveira, Mar 08 2025: (Start)
E.g.f.: exp(3*x)*(11*exp(8*x) - 3)/8.
a(n) = A139741(n+1)/8. (End)

Extensions

More terms from Michel Marcus, Mar 09 2025
Showing 1-2 of 2 results.

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

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