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.

A135343 a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).

Original entry on oeis.org

1, 3, 12, 51, 205, 820, 3277, 13107, 52428, 209715, 838861, 3355444, 13421773, 53687091, 214748364, 858993459, 3435973837, 13743895348, 54975581389, 219902325555, 879609302220, 3518437208883, 14073748835533, 56294995342132, 225179981368525, 900719925474099
Offset: 0

Views

Author

Paul Curtz, Dec 05 2007

Keywords

Comments

See A129339 comments. Third sequence after b(n) = 3*b(n-1) - 3*b(n-2) + 2*b(n-3) and c(n) = 3*c(n-1) - c(n-3) + 3*c(n-4). The first is for every sequence identical to its third differences. What characterizes the two others?

Links

Crossrefs

Cf. A129339, A135345.

Programs

  • Mathematica
    LinearRecurrence[{3,4,-1,3,4},{1,3,12,51,205},30] (* Harvey P. Dale, Jun 03 2013 *)
  • PARI
    Vec((1-x+4*x^3)/((1+x)*(1-4*x)*(1-x+x^2)) + O(x^30)) \\ Colin Barker, Oct 11 2016

Formula

a(n+1) - 4*a(n) = hexaperiodic -1, 0, 3, 1, 0, -3.
a(n) = (1/15)*( 3*4^(n+1) - 2*(-1)^n + 5*cos(Pi*n/3) - 5*sqrt(3)*cos(Pi*n/3) ). - Richard Choulet, Jan 04 2008
G.f.: (1-x+4*x^3) / ((1+x)*(1-4*x)*(1-x+x^2)). - Colin Barker, Oct 11 2016

Extensions

More terms from Harvey P. Dale, Jun 03 2013
Removed incorrect formula, Joerg Arndt, Oct 11 2016

A135450 a(n) = 3*a(n-1) + 4*a(n-2) - a(n-3) + 3*a(n-4) + 4*a(n-5).

Original entry on oeis.org

0, 0, 0, 1, 4, 16, 63, 252, 1008, 4033, 16132, 64528, 258111, 1032444, 4129776, 16519105, 66076420, 264305680, 1057222719, 4228890876, 16915563504, 67662254017, 270649016068, 1082596064272, 4330384257087, 17321537028348
Offset: 0

Views

Author

Paul Curtz, Dec 14 2007

Keywords

Links

Crossrefs

Cf. A135343, A135345.

Programs

  • Mathematica
    a = {0, 0, 0, 1, 4}; Do[AppendTo[a, 3*a[[ -1]] + 4*a[[ -2]] - a[[ -3]] + 3*a[[ -4]] + 4*a[[ -5]]], {25}]; a (* Stefan Steinerberger, Dec 31 2007 *)
LinearRecurrence[{3, 4, -1, 3, 4}, {0, 0, 0, 1, 4}, 25] (* G. C. Greubel, Oct 14 2016 *)
LinearRecurrence[{4,0,-1,4},{0,0,0,1},40] (* Harvey P. Dale, Jan 31 2021 *)
  • PARI
    a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; 4,-1,0,4]^n*[0;0;0;1])[1,1] \\ Charles R Greathouse IV, Oct 14 2016

Formula

a(n+1) - 4*a(n) = hexaperiodic 0, 0, 1, 0, 0, -1, A131531.
a(n) + a(n+3) = 1, 4, 16, 64 = 2^2n = A000302.
a(n) = (1/65)*4^n + (1/15)*(-1)^(n+1) + (2/39)*cos((Pi*n)/3) - (4*sqrt(3)/39) * sin((Pi*n)/3). Or, a(n) = (1/65)*(4^n + [ -1; -4; -16; 1; 4; 16]). - Richard Choulet, Dec 31 2007
O.g.f.: -x^3/[(4*x-1)*(1+x)*(x^2-x+1)]. - R. J. Mathar, Jan 07 2008

Extensions

More terms from Stefan Steinerberger, Dec 31 2007
Showing 1-2 of 2 results.

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