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 1-10 of 10 results.

A127210 a(n) = 3^n*Lucas(n), where Lucas = A000204.

Original entry on oeis.org

3, 27, 108, 567, 2673, 13122, 63423, 308367, 1495908, 7263027, 35252253, 171124002, 830642283, 4032042867, 19571909148, 95004113247, 461159522073, 2238515585442, 10865982454983, 52744587633927, 256027604996628, 1242784103695227, 6032600756055333, 29282859201423042
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A000204, A087131, A127211, A127212, A127213, A127214, A127215, A127216.

Programs

  • Magma
    [3^n*Lucas(n): n in [1..30]]; // Vincenzo Librandi, Aug 07 2015
  • Mathematica
    Table[3^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[3^n LucasL[n], {n, 25}] (* Vincenzo Librandi, Aug 07 2015 *)
  • PARI
    lucas(n) = fibonacci(n-1) + fibonacci(n+1);
vector(30, n, 3^n*lucas(n)) \\ Michel Marcus, Aug 07 2015

Formula

a(n) = Trace of matrix [({3,3},{3,0})^n] = 3^n * Trace of matrix [({1,1},{1,0})^n].
From R. J. Mathar, Oct 27 2008: (Start)
a(n) = 3*a(n-1) + 9*a(n-2).
G.f.: 3*x*(1 + 6*x)/(1 - 3*x - 9*x^2).
a(n) = 3*A099012(n) +18*A099012(n-1). (End)

Extensions

More terms from Michel Marcus, Aug 07 2015

A127212 a(n) = 5^n*Lucas(n), where Lucas = A000204.

Original entry on oeis.org

5, 75, 500, 4375, 34375, 281250, 2265625, 18359375, 148437500, 1201171875, 9716796875, 78613281250, 635986328125, 5145263671875, 41625976562500, 336761474609375, 2724456787109375, 22041320800781250, 178318023681640625
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A000204, A087131, A127210, A127211, A127213, A127214, A127215, A127216.

Programs

  • Magma
    [5^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017
  • Mathematica
    Table[5^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[5^n*LucasL[n], {n,1,50}] (* G. C. Greubel, Dec 18 2017 *)
LinearRecurrence[{5,25},{5,75},20] (* Harvey P. Dale, Jan 11 2024 *)
  • PARI
    x='x+O('x^30); Vec(-5*x*(10*x+1)/(25*x^2+5*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({5,5},{5,0})^n].
a(n) = 5^n * Trace of matrix [({1,1},{1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 5*a(n-1) + 25*a(n-2).
G.f.: -5*x*(10*x+1)/(25*x^2+5*x-1). (End)

A127213 a(n) = 6^n*Lucas(n), where Lucas = A000204.

Original entry on oeis.org

6, 108, 864, 9072, 85536, 839808, 8118144, 78941952, 765904896, 7437339648, 72196614144, 700923912192, 6804621582336, 66060990332928, 641332318961664, 6226189565755392, 60445100877152256, 586813429630107648
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A000204, A087131, A127210, A127211, A127212, A127214, A127215, A127216.

Programs

  • Magma
    [6^n*Lucas(n): n in [1..30]]; // G. C. Greubel, Dec 18 2017
  • Mathematica
    Table[6^n Tr[MatrixPower[{{1, 1}, {1, 0}}, x]], {x, 1, 20}]
Table[6^n*LucasL[n], {n,1,50}] (* G. C. Greubel, Dec 18 2017 *)
LinearRecurrence[{6,36},{6,108},20] (* Harvey P. Dale, Jan 20 2024 *)
  • PARI
    x='x+O('x^30); Vec(-6*x*(12*x+1)/(36*x^2+6*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({6,6},{6,0})^n].
a(n) = 6^n * Trace of matrix [({1,1},{1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 6*a(n-1) + 36*a(n-2).
G.f.: -6*x*(12*x+1)/(36*x^2+6*x-1). (End)

A127214 a(n) = 2^n*tribonacci(n) or (2^n)*A001644(n+1).

Original entry on oeis.org

2, 12, 56, 176, 672, 2496, 9088, 33536, 123392, 453632, 1669120, 6139904, 22585344, 83083264, 305627136, 1124270080, 4135714816, 15213527040, 55964073984, 205867974656, 757300461568, 2785785413632, 10247716470784, 37696978288640, 138671105769472
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A087131, A127210, A127211, A127212, A127213, A127215, A127216.

Programs

  • Magma
    I:=[2,12,56]; [n le 3 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 18 2017
  • Mathematica
    Table[Tr[MatrixPower[2*{{1, 1, 1}, {1, 0, 0}, {0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8}, {2, 12, 56}, 50] (* G. C. Greubel, Dec 18 2017 *)
  • PARI
    x='x+O('x^30); Vec(-2*x*(12*x^2+4*x+1)/(8*x^3+4*x^2+2*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({2,2,2},{2,0,0},{0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1},{1,0,0},{0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) + 8*a(n-3).
G.f.: -2*x*(12*x^2+4*x+1)/(8*x^3+4*x^2+2*x-1). (End)

Extensions

More terms from Colin Barker, Sep 02 2013

A127216 a(n) = 2^n*tetranacci(n) or (2^n)*A001648(n).

Original entry on oeis.org

2, 12, 56, 240, 832, 3264, 12672, 48896, 187904, 724992, 2795520, 10776576, 41541632, 160153600, 617414656, 2380201984, 9175957504, 35374497792, 136373075968, 525735034880, 2026773676032, 7813464064000, 30121872326656, 116123550875648, 447670682386432
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Examples

			a(8) = (2^8) * A001648(8) = 256 * 191  = 48896. - _Indranil Ghosh_, Feb 09 2017

Links

Crossrefs

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648.

Programs

  • Magma
    I:=[2,12,56,240]; [n le 4 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017
  • Mathematica
    Table[Tr[MatrixPower[2*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16}, {2, 12, 56, 240}, 50] (* G. C. Greubel, Dec 19 2017 *)
  • PARI
    x='x+O('x^30); Vec(-2*x*(32*x^3+12*x^2+4*x+1)/(16*x^4 +8*x^3 +4*x^2 +2*x -1)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) = Trace of matrix [({2,2,2,2},{2,0,0,0},{0,2,0,0},{0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 2*a(n-1) + 4*a(n-2) + 8*a(n-3) + 16*a(n-4).
G.f.: -2*x*(32*x^3+12*x^2+4*x+1) / (16*x^4+8*x^3+4*x^2+2*x-1). (End)

Extensions

More terms from Colin Barker, Sep 02 2013

A127215 a(n) = 3^n*tribonacci(n) or (3^n)*A001644(n+1).

Original entry on oeis.org

3, 27, 189, 891, 5103, 28431, 155277, 859491, 4743603, 26158707, 144374805, 796630059, 4395548511, 24254435799, 133832255589, 738466498755, 4074759563139, 22483948079115, 124063275771981, 684563868232731, 3777327684782127, 20842766314284447
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216.

Programs

  • Magma
    I:=[3,27,189]; [n le 3 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 18 2017
  • Mathematica
    Table[Tr[MatrixPower[3*{{1, 1, 1}, {1, 0, 0}, {0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27}, {3, 27, 189}, 50] (* G. C. Greubel, Dec 18 2017 *)
  • PARI
    x='x+O('x^30); Vec(-3*x*(27*x^2+6*x+1)/(27*x^3+9*x^2+3*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({3,3,3},{3,0,0},{0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1},{1,0,0},{0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3).
G.f.: -3*x*(27*x^2+6*x+1)/(27*x^3+9*x^2+3*x-1). (End)

Extensions

More terms from Colin Barker, Sep 02 2013

A127220 a(n) = 3^n*tetranacci(n) or (2^n)*A001648(n).

Original entry on oeis.org

3, 27, 189, 1215, 6318, 37179, 216513, 1253151, 7223661, 41806692, 241805655, 1398221271, 8084811933, 46753521975, 270362105694, 1563413859999, 9040715391141, 52279683047127, 302316992442837, 1748203962973380, 10109314209860523, 58458991419115875
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127221, A127222.

Programs

  • Magma
    I:=[3, 27, 189, 1215]; [n le 4 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4): n in [1..30]]; // G. C. Greubel, Dec 19 2017
  • Mathematica
    Table[Tr[MatrixPower[3*{{1, 1, 1, 1}, {1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81}, {3, 27, 189, 1215}, 50] (* G. C. Greubel, Dec 19 2017 *)
  • PARI
    x='x+O('x^30); Vec(-3*x*(108*x^3 +27*x^2 +6*x +1)/(81*x^4 +27*x^3 +9*x^2 +3*x -1)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) = Trace of matrix [({3,3,3,3},{3,0,0,0},{0,3,0,0},{0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1},{1,0,0,0},{0,1,0,0},{0,0,1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 3*a(n-1) + 9*a(n-2) + 27*a(n-3) + 81*a(n-4).
G.f.: -3*x*(108*x^3+27*x^2+6*x+1)/(81*x^4+27*x^3+9*x^2+3*x-1). (End)

Extensions

More terms from Colin Barker, Sep 02 2013

A127221 a(n) = 2^n*pentanacci(n) or (2^n)*A023424(n-1).

Original entry on oeis.org

2, 12, 56, 240, 992, 3648, 14464, 57088, 224768, 883712, 3471360, 13651968, 53682176, 211075072, 829915136, 3263102976, 12830244864, 50447253504, 198353354752, 779904614400, 3066503888896, 12057176965120, 47407572189184, 186401664532480, 732912043425792
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127222.
Cf. A023424, A074048.

Programs

  • Magma
    I:=[2, 12, 56, 240, 992]; [n le 5 select I[n] else 2*Self(n-1) + 4*Self(n-2) + 8*Self(n-3) + 16*Self(n-4) + 32*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017
  • Mathematica
    Table[Tr[MatrixPower[2*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{2, 4, 8, 16, 32}, {2, 12, 56, 240, 992}, 50] (* G. C. Greubel, Dec 19 2017 *)
  • PARI
    x='x+O('x^30); Vec(-2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) = Trace of matrix [({2,2,2,2,2},{2,0,0,0,0},{0,2,0,0,0},{0,0,2,0,0},{0,0,0,2,0})^n].
a(n) = 2^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -2*x*(1 +4*x +12*x^2 +32*x^3 +80*x^4)/(-1 +2*x +4*x^2 +8*x^3 +16*x^4 +32*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009; corrected by R. J. Mathar, Sep 16 2009
a(n) = 2*a(n-1)+4*a(n-2)+8*a(n-3)+16*a(n-4)+32*a(n-5). - Colin Barker, Sep 02 2013

Extensions

Definition corrected by R. J. Mathar, Sep 17 2009
More terms from Colin Barker, Sep 02 2013

A127222 a(n) = 3^n*pentanacci(n) or (3^n)*A023424(n-1).

Original entry on oeis.org

3, 27, 189, 1215, 7533, 41553, 247131, 1463103, 8640837, 50959287, 300264165, 1771292853, 10447598619, 61618989627, 363414767589, 2143339285311, 12641143135581, 74555586323649, 439717218548643, 2593383067853775, 15295369041550269, 90209719910309895
Offset: 1

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A087131, A127210, A127211, A127212, A127213, A127214, A127216, A001648, A127220, A127221.
Cf. A023424, A074048.

Programs

  • Magma
    I:=[3, 27, 189, 1215, 7533]; [n le 5 select I[n] else 3*Self(n-1) + 9*Self(n-2) + 27*Self(n-3) + 81*Self(n-4) + 243*Self(n-5): n in [1..30]]; // G. C. Greubel, Dec 19 2017
  • Mathematica
    Table[Tr[MatrixPower[3*{{1, 1, 1, 1, 1}, {1, 0, 0, 0, 0}, {0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}}, x]], {x, 1, 20}]
LinearRecurrence[{3, 9, 27, 81, 243}, {3, 27, 189, 1215, 7533}, 50] (* G. C. Greubel, Dec 19 2017 *)
  • PARI
    x='x+O('x^30); Vec(-3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5)) \\ G. C. Greubel, Dec 19 2017

Formula

a(n) = Trace of matrix [({3,3,3,3,3},{3,0,0,0,0},{0,3,0,0,0},{0,0,3,0,0},{0,0,0,3,0})^n].
a(n) = 3^n * Trace of matrix [({1,1,1,1,1},{1,0,0,0,0},{0,1,0,0,0},{0,0,1,0,0},{0,0,0,1,0})^n].
G.f.: -3*x*(1 +6*x +27*x^2 +108*x^3 +405*x^4)/(-1 +3*x +9*x^2 +27*x^3 +81*x^4 +243*x^5). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 28 2009
a(n) = 3*a(n-1)+9*a(n-2)+27*a(n-3)+81*a(n-4)+243*a(n-5). - Colin Barker, Sep 02 2013

Extensions

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
Definition corrected by R. J. Mathar, Sep 17 2009
More terms from Colin Barker, Sep 02 2013

A371986 Product of Lucas and Catalan numbers: a(n) = A000032(n)*A000108(n).

Original entry on oeis.org

2, 1, 6, 20, 98, 462, 2376, 12441, 67210, 369512, 2065908, 11698414, 66979864, 387050900, 2254552920, 13223768580, 78034377690, 462961545090, 2759796408600, 16522143563310, 99295449593340, 598836351581520, 3622983967834920, 21982916983078350, 133739841802846968
Offset: 0

Views

Author

Vladimir Kruchinin, Apr 15 2024

Keywords

Crossrefs

Cf. A000032, A000108, A098614, A119694, A127211, A216541.

Programs

  • Maple
    From Peter Luschny, Apr 15 2024: (Start)
a := n -> ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * GAMMA(n + 1/2) / (sqrt(Pi) * GAMMA(n + 2)): seq(simplify(a(n)), n = 0..24);
# With g.f.:
assume(x>0); f := sqrt(1 - 4*x*(4*x + 1)):
gf := (sqrt(1 + f - 2*x) + sqrt(5)*sqrt(1 - f - 2*x) - sqrt(2))/(sqrt(8)*x):
ser := series(gf, x, 26): seq(simplify(coeff(ser, x, n)), n = 0..24);
# Recurrence:
a := proc(n) option remember: if n < 2 then return [2, 1][n + 1] fi;
2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) end:
seq(a(n), n=0..24);  (End)
  • Mathematica
    a[n_] := a[n] = Switch[n, 0, 2, 1, 1, _, 2*(2n - 1)*(n*a[n - 1] + (4n - 6)*a[n - 2])/(n*(n + 1))];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Jun 17 2024, after Peter Luschny *)
  • Python
    def A371986_gen(): # generator of terms
    a, b, n = 2, 1, 2
    while True:
        yield a
        a, b = b, (4*n - 2)*(n*b + (4*n - 6)*a) // (n*n + n)
        n += 1
def A371986_list(len):
    it =  A371986_gen()
    return [next(it) for _ in range(len)]
print(A371986_list(25))  # Peter Luschny, Apr 15 2024

Formula

G.f.: (5*sqrt(-sqrt(-16*x^2 - 4*x+1) - 2*x+1)) / (2*sqrt(10)*x) - (1 - sqrt(sqrt( -16*x^2 - 4*x+1) - 2*x + 1) / sqrt(2)) / (2*x).
E.g.f.: exp(x-sqrt(5)*x)*(BesselI(0, x-sqrt(5)*x) - BesselI(1, x-sqrt(5)*x) + exp(2*sqrt(5)*x) * (BesselI(0, x+sqrt(5)*x) - BesselI(1, x+sqrt(5)*x))). - Stefano Spezia, Apr 15 2024
From Peter Luschny, Apr 15 2024: (Start)
a(n) = 2*(2*n - 1)*(n*a(n - 1) + (4*n - 6)*a(n - 2)) / (n*(n + 1)) for n >= 2.
a(n) = ((2 - 2*sqrt(5))^n + (2 + 2*sqrt(5))^n) * Gamma(n + 1/2) / (sqrt(Pi) * Gamma(n + 2)).
a(n) ~ (2 + 2*sqrt(5))^n / (n*(n*Pi)^(1/2)). (End)
Showing 1-10 of 10 results.

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