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.

A135345 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, 4, 13, 51, 204, 819, 3277, 13108, 52429, 209715, 838860, 3355443, 13421773, 53687092, 214748365, 858993459, 3435973836, 13743895347, 54975581389, 219902325556, 879609302221, 3518437208883, 14073748835532, 56294995342131, 225179981368525, 900719925474100, 3602879701896397, 14411518807585587
Offset: 0

Views

Author

Paul Curtz, Dec 06 2007

Keywords

Comments

Companion to A135343.

Links

Crossrefs

Cf. A135343.

Programs

  • Mathematica
    LinearRecurrence[{3,4,-1,3,4},{1,4,13,51,204}, 25] (* G. C. Greubel, Oct 10 2016 *)
  • PARI
    Vec((1-3*x^2)/((1+x)*(1-4*x)*(1-x+x^2)) + O(x^30)) \\ Colin Barker, Oct 11 2016

Formula

4*a(n) - a(n+1) = hexaperiodic 0, 3, 1, 0, -3, -1.
a(n) = (4^(n+1)/5)-(2/15)*(-1)^n+(1/3)*cos(Pi*n/3)+(sqrt(3)/3)*cos(Pi*n/3). - Richard Choulet, Jan 04 2008
G.f.: ( -2*(3 + sqrt(3)) + (3 + 7*sqrt(3))*x + (9 + 5*sqrt(3))*x^2 -
4*(3 + sqrt(3))*x^3)/( 6*(-1 + 4*x - x^3 + 4*x^4) ). - G. C. Greubel, Oct 10 2016
G.f.: (1-3*x^2) / ((1+x)*(1-4*x)*(1-x+x^2)). - Colin Barker, Oct 11 2016

Extensions

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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