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.

A249992 Expansion of 1/((1+x)*(1+2*x)*(1-3*x)).

Original entry on oeis.org

1, 0, 7, 6, 49, 84, 379, 882, 3157, 8448, 27391, 78078, 242425, 710892, 2165443, 6430794, 19423453, 58008216, 174548935, 522598230, 1569891841, 4705481220, 14124832267, 42357719586, 127106713189, 381253030704, 1143893309839, 3431411494062, 10294771353097
Offset: 0

Views

Author

Alex Ratushnyak, Dec 27 2014

Keywords

Links

Crossrefs

Cf. A249993.
Cf. A000392: expansion of x^3/((1-x)*(1-2*x)*(1-3*x)).
Cf. A091002: expansion of x^2/((1-x)*(1+2*x)*(1-3*x)).
Cf. A094705: expansion of x/((1+x)*(1-2*x)*(1-3*x)).

Programs

  • Magma
    [(3^(n+2) + (-1)^n*(2^(n+4) - 5))/20: n in [0..50]]; // G. C. Greubel, Jul 21 2022
  • Maple
    seq((9/20)*3^n+(4/5)*(-2)^n-(1/4)*(-1)^n, n=0 .. 100); # Robert Israel, Dec 28 2014
  • Mathematica
    LinearRecurrence[{0, 7, 6}, {1, 0, 7}, 29] (* Jean-François Alcover, Oct 05 2017 *)
CoefficientList[Series[1/((1+x)(1+2x)(1-3x)),{x,0,30}],x] (* Harvey P. Dale, May 26 2020 *)
  • PARI
    Vec(1/((1+x)*(1+2*x)*(1-3*x)) + O(x^50)) \\ Michel Marcus, Dec 28 2014
  • SageMath
    [(3^(n+2) +(-1)^n*(2^(n+4) -5))/20 for n in (0..50)] # G. C. Greubel, Jul 21 2022

Formula

G.f.: 1/((1+x)*(1+2*x)*(1-3*x)).
a(n) = ( 3^(n+2) + (2^(n+4) - 5)*(-1)^n )/20. - Colin Barker, Dec 28 2014
a(n) = 7*a(n-2) + 6*a(n-3). - Colin Barker, Dec 28 2014
E.g.f.: (9/20)*exp(3*x) + (4/5)*exp(-2*x) - (1/4)*exp(-x). - Robert Israel, Dec 28 2014

A249994 Expansion of 1/((1-2*x)*(1+3*x)*(1-4*x)).

Original entry on oeis.org

1, 3, 19, 63, 307, 1095, 4843, 18111, 76483, 294327, 1213147, 4747119, 19308979, 76282599, 308006731, 1223430687, 4919576995, 19600876311, 78636062395, 313847102415, 1257480899731, 5023648225863, 20113423216939, 80397210315903, 321758305696387, 1286524863041655
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016269 for the expansion of 1/((1-2*x)*(1-3*x)*(1-4*x)).
Cf. A249992, A249993, A249995, A249996.

Programs

  • Magma
    [(5*2^(2*n+3) -7*2^(n+1) +(-1)^n*3^(n+2))/35: n in [0..40]]; // G. C. Greubel, Oct 10 2022
  • Mathematica
    LinearRecurrence[{3,10,-24}, {1,3,19}, 41] (* G. C. Greubel, Oct 10 2022 *)
  • PARI
    Vec(1/((2*x-1)*(3*x+1)*(4*x-1)) + O(x^100)) \\ Colin Barker, Dec 29 2014
  • SageMath
    [(5*2^(2*n+3) -7*2^(n+1) +(-1)^n*3^(n+2))/35 for n in range(41)] # G. C. Greubel, Oct 10 2022

Formula

G.f.: 1/((1-2*x)*(1+3*x)*(1-4*x)).
a(n) = (5*2^(2*n+3) - 7*2^(n+1) + (-1)^n*3^(n+2))/35. - Colin Barker, Dec 29 2014
a(n) = 3*a(n-1) + 10*a(n-2) - 24*a(n-3). - Colin Barker, Dec 29 2014
E.g.f.: (1/35)*(9*exp(-3*x) - 14*exp(2*x) + 40*exp(4*x)). - G. C. Greubel, Oct 10 2022

A249995 Expansion of 1/((1+2*x)*(1-3*x)*(1-4*x)).

Original entry on oeis.org

1, 5, 27, 121, 539, 2289, 9619, 39737, 162987, 663553, 2690051, 10865673, 43783195, 176086097, 707220723, 2837479129, 11375770763, 45580514721, 182554616035, 730915611305, 2925754935291, 11709295114225, 46856010770387, 187480525633401, 750091566966379, 3000874627609409
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016269 for the expansion of 1/((1-2*x)*(1-3*x)*(1-4*x)).
Cf. A249992, A249993, A249994, A249996.

Programs

  • Magma
    [(5*2^(2*n+3) +(-1)^n*2^(n+1) -3^(n+3))/15: n in [0..40]]; // G. C. Greubel, Oct 10 2022
  • Mathematica
    LinearRecurrence[{5,2,-24}, {1,5,27}, 41] (* G. C. Greubel, Oct 10 2022 *)
CoefficientList[Series[1/((1+2x)(1-3x)(1-4x)),{x,0,40}],x] (* Harvey P. Dale, Oct 28 2022 *)
  • PARI
    Vec(1/((1+2*x)*(1-3*x)*(1-4*x)) + O(x^50)) \\ Michel Marcus, Dec 29 2014
  • SageMath
    [(5*2^(2*n+3) +(-1)^n*2^(n+1) -3^(n+3))/15 for n in range(41)] # G. C. Greubel, Oct 10 2022

Formula

G.f.: 1/((1+2*x)*(1-3*x)*(1-4*x)).
a(n) = ((-1)^n*2^(n+1) + 5*2^(2*n+3) - 3^(n+3))/15. - Colin Barker, Dec 29 2014
a(n) = 5*a(n-1) + 2*a(n-2) - 24*a(n-3). - Colin Barker, Dec 29 2014
E.g.f.: (1/15)*(2*exp(-2*x) - 27*exp(3*x) + 40*exp(4*x)). - G. C. Greubel, Oct 10 2022

A249996 Expansion of 1/((1+2*x)*(1+3*x)*(1-4*x)).

Original entry on oeis.org

1, -1, 15, -5, 191, 99, 2455, 3515, 33231, 74899, 474695, 1371435, 7071871, 23520899, 108399735, 390617755, 1691480111, 6378762099, 26676785575, 103221406475, 423343881951, 1661998662499, 6742129440215, 26686105001595, 107591675061391, 427824901526099, 1718925069371655
Offset: 0

Views

Author

Alex Ratushnyak, Dec 28 2014

Keywords

Links

Crossrefs

Cf. A016269: expansion of 1/((1-2*x)*(1-3*x)*(1-4*x)).
Cf. A249992, A249993, A249994, A249995.

Programs

  • Magma
    [(2^(2*n+3) +(-1)^n*(3^(n+3) -7*2^(n+1)))/21: n in [0..40]]; // G. C. Greubel, Oct 11 2022
  • Mathematica
    LinearRecurrence[{-1,14,24}, {1,-1,15}, 41] (* G. C. Greubel, Oct 11 2022 *)
  • PARI
    Vec(1/((1+2*x)*(1+3*x)*(1-4*x)) + O(x^50)) \\ Michel Marcus, Dec 29 2014
  • SageMath
    [(2^(2*n+3) +(-1)^n*(3^(n+3) -7*2^(n+1)))/21 for n in range(41)] # G. C. Greubel, Oct 11 2022

Formula

G.f.: 1 / ((1+2*x)*(1+3*x)*(1-4*x)).
a(n) = ( 2^(3+2*n) + (3^(3+n)-7*2^(1+n))*(-1)^n )/21. - Colin Barker, Dec 29 2014
a(n) = -a(n-1) + 14*a(n-2) + 24*a(n-3). - Colin Barker, Dec 29 2014
E.g.f.: (1/21)*(27*exp(-3*x) - 14*exp(-2*x) + 8*exp(4*x)). - G. C. Greubel, Oct 11 2022
Showing 1-4 of 4 results.

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