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.

A132868 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), with initial values 1,3,7,20.

Original entry on oeis.org

1, 3, 7, 20, 60, 182, 547, 1641, 4921, 14762, 44286, 132860, 398581, 1195743, 3587227, 10761680, 32285040, 96855122, 290565367, 871696101, 2615088301, 7845264902, 23535794706, 70607384120, 211822152361, 635466457083, 1906399371247, 5719198113740
Offset: 0

Views

Author

Paul Curtz, Nov 22 2007

Keywords

Links

Crossrefs

Cf. A129339, A132951.

Programs

  • Mathematica
    LinearRecurrence[{3,0,-1,3},{1,3,7,20},50] (* Harvey P. Dale, Jan 21 2012 *)

Formula

4*a(n) = 3^(n+1) + A132951(n).
O.g.f.: (-1+2*x^2)/((3*x-1)*(x+1)*(x^2-x+1)) = -(3/4)/(3*x-1)-(1/12)/(x+1)+(1/3)*(x+1)/(x^2-x+1). - R. J. Mathar, Nov 28 2007
a(n) = (1/12)*(3^(n+2) - 4*cos((n+1)*Pi/3) + cos((n+1)*Pi) + 4*sqrt(3) * sin(((n+1)*Pi)/3) + I*sin((n+1)*Pi)). - Harvey P. Dale, Jan 21 2012
12*a(n) = -(-1)^n +3^(n+2) +4*A057079(n). - R. J. Mathar, Oct 03 2021

A132353 a(n) = 3*a(n-1) - a(n-3) + 3*a(n-4), starting with 1, 2, 6, 20.

Original entry on oeis.org

1, 2, 6, 20, 61, 183, 547, 1640, 4920, 14762, 44287, 132861, 398581, 1195742, 3587226, 10761680, 32285041, 96855123, 290565367, 871696100, 2615088300, 7845264902, 23535794707, 70607384121, 211822152361, 635466457082
Offset: 0

Views

Author

Paul Curtz, Nov 24 2007

Keywords

Comments

A132868(n) - a(n) = A128834(n) (discovered in 1995).

Links

Crossrefs

Cf. A129339.

Programs

  • Magma
    I:=[1,2,6,20]; [n le 4 select I[n] else 3*Self(n-1) - Self(n-3) + 3*Self(n-4): n in [1..30]]; // G. C. Greubel, Jan 15 2018
  • Mathematica
    LinearRecurrence[{3, 0, -1, 3}, {1, 2, 6, 20}, 50] (* G. C. Greubel, Jan 15 2018 *)
nxt[{a_,b_,c_,d_}]:={b,c,d,3d-b+3a}; NestList[nxt,{1,2,6,20},50][[;;,1]] (* Harvey P. Dale, Feb 17 2025 *)
  • PARI
    x='x+O('x^30); Vec((1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1))) \\ G. C. Greubel, Jan 15 2018

Formula

Also a(n) - 3^(n+1) = hexaperiodic 1, -1, -3, -1, 1, 3; cf. A132951.
From R. J. Mathar, Apr 04 2008: (Start)
O.g.f.: (1-x+3*x^3)/((1-3*x)*(1+x)*(x^2-x+1)).
a(n) = -(-1)^n/12 + 3^(n+1)/4 + A057079(n+2)/3. (End)

Extensions

More terms from R. J. Mathar, Apr 04 2008
Showing 1-2 of 2 results.

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