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.

A127211 a(n) = 4^n*Lucas(n), where Lucas = A000032.

Original entry on oeis.org

2, 4, 48, 256, 1792, 11264, 73728, 475136, 3080192, 19922944, 128974848, 834666496, 5402263552, 34963718144, 226291089408, 1464583847936, 9478992822272, 61349312856064, 397061136580608, 2569833552019456, 16632312393367552, 107646586405781504, 696703343917006848
Offset: 0

Views

Author

Artur Jasinski, Jan 09 2007

Keywords

Links

Crossrefs

Cf. A000032, A000204, A087131, A087131, A127210, A127212, A127213, A127214, A127215, A127216.

Programs

  • Magma
    [4^n*Lucas(n): n in [0..30]]; // G. C. Greubel, Dec 18 2017
  • Maple
    a:= n-> 4^n*(<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1]:
seq(a(n), n=0..22);  # Alois P. Heinz, Apr 15 2024
  • Mathematica
    Table[4^n Tr[MatrixPower[{{1, 1}, {1, 0}}, n]], {n, 0, 20}]
Table[4^n*LucasL[n], {n, 0, 50}] (* G. C. Greubel, Dec 18 2017 *)
  • PARI
    my(x='x + O('x^30)); Vec(-4*x*(8*x+1)/(16*x^2+4*x-1)) \\ G. C. Greubel, Dec 18 2017

Formula

a(n) = Trace of matrix [({4,4},{4,0})^n].
a(n) = 4^n * Trace of matrix [({1,1},{1,0})^n].
From Colin Barker, Sep 02 2013: (Start)
a(n) = 4*a(n-1) + 16*a(n-2).
G.f.: 2*x*(2*x-1)/(16*x^2+4*x-1). (End)
From Peter Luschny, Apr 15 2024: (Start)
a(n) = 2^n*((1 - sqrt(5))^n + (1 + sqrt(5))^n).
a(n) = 4^n*(Fibonacci(n+1) + Fibonacci(n-1)). (End)
a(n) = 2^n*A087131(n). - Michel Marcus, Apr 15 2024

Extensions

a(0)=2 prepended by Alois P. Heinz, Apr 15 2024

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
Showing 1-5 of 5 results.

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