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 1-4 of 4 results.

A159329 Transform of the finite sequence (1, 0, -1) by the T_{1,1} transformation (see link).

Original entry on oeis.org

2, 4, 9, 23, 54, 125, 290, 674, 1567, 3643, 8469, 19688, 45769, 106400, 247350, 575019, 1336757, 3107583, 7224254, 16794353, 39042134, 90761950, 210995935, 490506039, 1140288197, 2650848448, 6162474989, 14326016268, 33303947274
Offset: 0

Views

Author

Richard Choulet, Apr 10 2009

Keywords

Links

Crossrefs

Cf. A159328.

Programs

  • Magma
    I:=[9, 23, 54]; [2,4] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 26 2018
  • Mathematica
    Join[{2,4}, LinearRecurrence[{3, -2, 1}, {9, 23, 54}, 50]] (* G. C. Greubel, Jun 26 2018 *)
  • PARI
    x='x+O('x^30); Vec(2+4*x -x^2*(9-4*x+3*x^2)/(-1+3*x-2*x^2+x^3)) \\ G. C. Greubel, Jun 26 2018

Formula

O.g.f.: 2+4*x -x^2*(9-4*x+3*x^2) / ( -1+3*x-2*x^2+x^3 ).
a(0)=2, a(1)=4, a(2)=9, a(3)=23, a(4)=54 and for n>=2 a(n+3)=3*a(n+2)-2*a(n+1)+a(n).

A159330 Transform of the finite sequence (1, 0, -1, 0, 1) by the T_{1,1} transformation (see link).

Original entry on oeis.org

2, 4, 9, 23, 55, 126, 292, 679, 1579, 3671, 8534, 19839, 46120, 107216, 249247, 579429, 1347009, 3131416, 7279659, 16923154, 39341560, 91458031, 212614127, 494267879, 1149033414, 2671178611, 6209736884, 14435886844, 33559365375, 78016059321, 181365334057
Offset: 0

Views

Author

Richard Choulet, Apr 10 2009

Keywords

Links

Crossrefs

Cf. A159328, A159329.

Programs

  • Magma
    I:=[55, 126, 292]; [2, 4, 9, 23] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 26 2018
  • Mathematica
    Join[{2, 4, 9, 23}, LinearRecurrence[{3, -2, 1}, {55, 126, 292}, 47]] (* G. C. Greubel, Jun 26 2018 *)
  • PARI
    my(z='z+O('z^31)); Vec(((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4) + z/(1-3*z+2*z^2-z^3) + (1-z+z^2)/(1-3*z+2*z^2-z^3)) \\ G. C. Greubel, Jun 26 2018

Formula

O.g.f.: f(z) = ((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4) + z/(1-3*z+2*z^2-z^3) + (1-z+z^2)/(1-3*z+2*z^2-z^3).
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) for n >= 7, with a(0)=2, a(1)=4, a(2)=9, a(3)=23, a(4)=55, a(5)=126, a(6)=292.

A159331 Transform of the finite sequence (1, 0, -1, 0, 1, 0, -1) by the T_{1,1} transform (see link).

Original entry on oeis.org

2, 4, 9, 23, 55, 126, 293, 680, 1581, 3676, 8546, 19867, 46185, 107367, 249598, 580245, 1348906, 3135826, 7289911, 16946987, 39396965, 91586832, 212913553, 494963960, 1150651606, 2674940451, 6218482101, 14456217007, 33606627270
Offset: 0

Views

Author

Richard Choulet, Apr 10 2009

Keywords

Links

Crossrefs

Cf. A159328, A159329, A159330.

Programs

  • Magma
    I:=[293, 680, 1581]; [2, 4, 9, 23, 55, 126] cat [n le 3 select I[n] else 3*Self(n-1) - 2*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Jun 26 2018
  • Mathematica
    Join[{2, 4, 9, 23, 55, 126}, LinearRecurrence[{3, -2, 1}, {293, 680, 1581}, 45]] (* G. C. Greubel, Jun 26 2018 *)
  • PARI
    z='z+O('z^30); Vec(((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4+z^6) + (z/(1-3*z+2*z^2-z^3)) + ((1-z+z^2)/(1-3*z+2*z^2-z^3))) \\ G. C. Greubel, Jun 26 2018

Formula

O.g.f.: f(z) = ((1-z)^2/(1-3*z+2*z^2-z^3))*(1-z^2+z^4+z^6) + (z/(1-3*z+2*z^2-z^3)) + ((1-z+z^2)/(1-3*z+2*z^2-z^3)).
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) for n >= 9, with a(0)=2, a(1)=4, a(2)=9, a(3)=23, a(4)=55, a(5)=126, a(6)=293, a(7)=680, a(8)=1581.

A159334 Transform of A056594 by the T_{1,1} transformation (see link).

Original entry on oeis.org

2, 4, 9, 23, 55, 126, 291, 678, 1578, 3667, 8523, 19815, 46066, 107089, 248950, 578740, 1345409, 3127695, 7271007, 16903042, 39294807, 91349342, 212361454, 493680487, 1147667895, 2668004163, 6202357186, 14418731129, 33519483178
Offset: 0

Views

Author

Richard Choulet, Apr 10 2009

Keywords

Links

Crossrefs

Cf. A159328, A159329, A159330, A159331.

Programs

  • Magma
    I:=[291, 678, 1578, 3667, 8523]; [2, 4, 9, 23, 55, 126] cat [n le 5 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 4*Self(n-3) -2*Self(n-4) +Self(n-5): n in [1..50]]; // G. C. Greubel, Jun 25 2018
  • Mathematica
    Join[{2, 4, 9, 23, 55, 126}, LinearRecurrence[{3, -3, 4, -2, 1}, {291, 678, 1578, 3667, 8523}, 45]] (* G. C. Greubel, Jun 25 2018 *)
  • PARI
    x='x+O('x^50); Vec(-(2-2*x+3*x^2+x^4)/((x^2+1)*(x^3-2*x^2+3*x-1))) \\ G. C. Greubel, Jun 25 2018

Formula

O.g.f.: -(2-2*x+3*x^2+x^4)/((x^2+1)*(x^3-2*x^2+3*x-1)).
for n>=0 a(n+5)=3*a(n+4)-3*a(n+3)+4*a(n+2)-2*a(n+1)+a(n)
Showing 1-4 of 4 results.

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