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 1-5 of 5 results.

A099783 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k)*3^(n-2*k).

Original entry on oeis.org

1, 3, 9, 30, 108, 405, 1548, 5967, 23085, 89451, 346842, 1345248, 5218263, 20242872, 78528609, 304640595, 1181814705, 4584708702, 17785841652, 68998115709, 267670245492, 1038395956527, 4028337876861, 15627474388899, 60624993311226
Offset: 0

Views

Author

Paul Barry, Oct 26 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).

Links

Crossrefs

Cf. A099780, A099781, A099782, A099784, A099785, A099786, A099787.

Programs

  • GAP
    a:=[1,3,9];; for n in [4..30] do a[n]:=6*a[n-1]-9*a[n-2]+3*a[n-3]; od; a; # G. C. Greubel, Sep 04 2019
  • Magma
    I:=[1,3,9]; [n le 3 select I[n] else 6*Self(n-1) - 9*Self(n-2) + 3*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019
  • Maple
    seq(coeff(series((1-3*x)/((1-3*x)^2 - 3*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
  • Mathematica
    LinearRecurrence[{6,-9,3}, {1,3,9}, 30] (* G. C. Greubel, Sep 04 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-3*x)/((1-3*x)^2 - 3*x^3)) \\ G. C. Greubel, Sep 04 2019
  • Sage
    def A099783_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/((1-3*x)^2 - 3*x^3)).list()
A099783_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-3*x)/((1-3*x)^2 - 3*x^3).
a(n) = 6*a(n-1) - 9*a(n-2) + 3*a(n-3).

A099785 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^(n-3*k).

Original entry on oeis.org

1, 2, 4, 8, 18, 48, 144, 448, 1380, 4152, 12224, 35456, 102024, 292768, 840416, 2416384, 6959504, 20069280, 57913536, 167158656, 482462752, 1392319488, 4017460224, 11590946816, 33439639616, 96470796672, 278311599616
Offset: 0

Views

Author

Paul Barry, Oct 26 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

Links

Crossrefs

Cf. A003522, A097119.
Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

Programs

  • GAP
    a:=[1,2,4,8];; for n in [5..30] do a[n]:=6*a[n-1] -12*a[n-2] + 8*a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, Sep 04 2019
  • Magma
    I:=[1,2,4,8]; [n le 4 select I[n] else 6*Self(n-1) - 12*Self(n-2) + 8*Self(n-3) + 2*Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019
  • Maple
    seq(coeff(series((1-2*x)^2/((1-2*x)^3 - 2*x^4), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 04 2019
  • Mathematica
    Table[Sum[Binomial[n-k,3k]2^(n-3k),{k,0,Floor[n/4]}],{n,0,30}] (* or *) LinearRecurrence[{6,-12,8,2},{1,2,4,8},30] (* Harvey P. Dale, Apr 01 2012 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-2*x)^2/((1-2*x)^3 - 2*x^4)) \\ G. C. Greubel, Sep 04 2019
  • Sage
    def A099785_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-2*x)^2/((1-2*x)^3 - 2*x^4)).list()
A099785_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-2*x)^2/((1-2*x)^3 - 2*x^4).
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3) + 2*a(n-4).

A099786 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k)*3^(n-4*k).

Original entry on oeis.org

1, 3, 9, 27, 82, 255, 819, 2727, 9397, 33312, 120537, 441855, 1631017, 6036879, 22345074, 82589247, 304612975, 1120960983, 4116353265, 15088372416, 55224373105, 201895801851, 737506551321, 2692518758163, 9826402960882
Offset: 0

Views

Author

Paul Barry, Oct 26 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

Links

Crossrefs

Cf. A003522, A097119.
Cf. A099780, A099781, A099782, A099783, A099784, A099785, A099787.

Programs

  • GAP
    a:=[1,3,9,27];; for n in [5..30] do a[n]:=9*a[n-1]-27*a[n-2] + 27*a[n-3] +a[n-4]; od; a; # G. C. Greubel, Sep 04 2019
  • Magma
    I:=[1,3,9,27]; [n le 4 select I[n] else 9*Self(n-1) - 27*Self(n-2) + 27*Self(n-3) +Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019
  • Maple
    seq(coeff(series((1-3*x)^2/((1-3*x)^3 - x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
  • Mathematica
    LinearRecurrence[{9,-27,27,1},{1,3,9,27},40] (* or *) CoefficientList[ Series[-((1-3 x)^2/(x (x (x (x+27)-27)+9)-1)),{x,0,40}],x] (* Harvey P. Dale, Jun 06 2011 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - x^4)) \\ G. C. Greubel, Sep 04 2019
  • Sage
    def A099786_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)^2/((1-3*x)^3 - x^4)).list()
A099786_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-3*x)^2/((1-3*x)^3 - x^4).
a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) + a(n-4).

A099787 a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * 2^k * 3^(n-4*k).

Original entry on oeis.org

1, 3, 9, 27, 83, 267, 909, 3267, 12235, 46983, 182529, 711099, 2764619, 10704147, 41257341, 158371011, 605932099, 2312728095, 8812918161, 33549513579, 127652354627, 485608571547, 1847326271949, 7028217617859, 26742885359131
Offset: 0

Views

Author

Paul Barry, Oct 26 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/4)} C(n-k,3*k) * u^k * v^(n-4*k) has g.f. (1-v*x)^2/((1-v*x)^3 - u*x^4) and satisfies the recurrence a(n) = 3*v*a(n-1) - 3*v^2*a(n-2) + v^3*a(n-3) + u*a(n-4).

Links

Crossrefs

Cf. A099780, A099781, A099783, A099784, A099785, A099786.

Programs

  • GAP
    a:=[1,3,9,27];; for n in [5..30] do a[n]:=9*a[n-1]-27*a[n-2] + 27*a[n-3] +2*a[n-4]; od; a; # G. C. Greubel, Sep 04 2019
  • Magma
    I:=[1,3,9,27]; [n le 4 select I[n] else 9*Self(n-1) - 27*Self(n-2) + 27*Self(n-3) + 2*Self(n-4): n in [1..30]]; // G. C. Greubel, Sep 04 2019
  • Maple
    seq(coeff(series((1-3*x)^2/((1-3*x)^3 - 2*x^4), x, n+1), x, n), n = 0..30); # G. C. Greubel, Sep 04 2019
  • Mathematica
    LinearRecurrence[{9,-27,27,2}, {1,3,9,27}, 30] (* G. C. Greubel, Sep 04 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-3*x)^2/((1-3*x)^3 - 2*x^4)) \\ G. C. Greubel, Sep 04 2019
  • Sage
    def A099787_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)^2/((1-3*x)^3 - 2*x^4)).list()
A099787_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-3*x)^2/((1-3*x)^3 - 2*x^4).
a(n) = 9*a(n-1) - 27*a(n-2) + 27*a(n-3) + 2*a(n-4).

A099782 a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * 2^k * 4^(n-3*k).

Original entry on oeis.org

1, 4, 16, 66, 280, 1216, 5380, 24144, 109504, 500488, 2300128, 10612224, 49096720, 227578432, 1056304384, 4907373600, 22813275520, 106100835328, 493609021504, 2296885357824, 10689540189184, 49753373831296, 231588118339072
Offset: 0

Views

Author

Paul Barry, Oct 26 2004

Keywords

Comments

In general a(n) = Sum_{k=0..floor(n/3)} C(n-k,2*k) * u^k * v^(n-3*k) has g.f. (1-v*x)/((1-v*x)^2 - u*x^2) and satisfies the recurrence a(n) = 2*u*v*a(n-1) - v^2*a(n-2) + u*a(n-3).

Links

Crossrefs

Cf. A099780, A099781, A099783, A099784, A099785, A099786, A099787.

Programs

  • GAP
    a:=[1,4,16];; for n in [4..30] do a[n]:=8*a[n-1]-16*a[n-2] + 2*a[n-3]; od; a; # G. C. Greubel, Sep 04 2019
  • Magma
    I:=[1,4,16]; [n le 3 select I[n] else 8*Self(n-1) - 16*Self(n-2) + 2*Self(n-3): n in [1..30]]; // G. C. Greubel, Sep 04 2019
  • Maple
    seq(coeff(series((1-4*x)/((1-4*x)^2 - 2*x^3), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Sep 04 2019
  • Mathematica
    LinearRecurrence[{8,-16,2}, {1,4,16}, 30] (* G. C. Greubel, Sep 04 2019 *)
  • PARI
    my(x='x+O('x^30)); Vec((1-4*x)/((1-4*x)^2 - 2*x^3)) \\ G. C. Greubel, Sep 04 2019
  • Sage
    def A099782_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-4*x)/((1-4*x)^2 - 2*x^3)).list()
A099782_list(30) # G. C. Greubel, Sep 04 2019

Formula

G.f.: (1-4*x)/((1-4*x)^2 - 2*x^3).
a(n) = 8*a(n-1) - 16*a(n-2) + 2*a(n-3).
Showing 1-5 of 5 results.

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