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 1-3 of 3 results.

A163604 a(n) = ((3+2*sqrt(2))*(4+sqrt(2))^n + (3-2*sqrt(2))*(4-sqrt(2))^n)/2.

Original entry on oeis.org

3, 16, 86, 464, 2508, 13568, 73432, 397504, 2151984, 11650816, 63078752, 341518592, 1849046208, 10011109376, 54202228096, 293462293504, 1588867154688, 8602465128448, 46575580861952, 252170135097344, 1365302948711424, 7392041698328576, 40022092304668672
Offset: 0

Views

Author

Al Hakanson (hawkuu(AT)gmail.com), Aug 01 2009

Keywords

Comments

Binomial transform of A163606. Inverse binomial transform of A163605.

Links

Crossrefs

Cf. A163606, A163605.

Programs

  • Magma
    Z:= PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((3+2*r)*(4+r)^n+(3-2*r)*(4-r)^n)/2: n in [0..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Aug 07 2009
  • Mathematica
    LinearRecurrence[{8, -14}, {3, 16}, 50] (* G. C. Greubel, Jul 29 2017 *)
  • PARI
    x='x+O('x^50); Vec((3-8*x)/(1-8*x+14*x^2)) \\ G. C. Greubel, Jul 29 2017

Formula

a(n) = 8*a(n-1)-14*a(n-2) for n > 1; a(0) = 3, a(1) = 16.
G.f.: (3-8*x)/(1-8*x+14*x^2).
E.g.f.: exp(4*x)*( 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x) ). - G. C. Greubel, Jul 29 2017

Extensions

Edited and extended beyond a(5) by Klaus Brockhaus and R. J. Mathar, Aug 07 2009

A163978 a(n) = 2*a(n-2) for n > 2; a(1) = 3, a(2) = 4.

Original entry on oeis.org

3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024, 1536, 2048, 3072, 4096, 6144, 8192, 12288, 16384, 24576, 32768, 49152, 65536, 98304, 131072, 196608, 262144, 393216, 524288, 786432, 1048576, 1572864, 2097152, 3145728
Offset: 1

Views

Author

Klaus Brockhaus, Aug 07 2009

Keywords

Comments

Interleaving of A007283 and A000079 without initial terms 1 and 2.
Equals A029744 without first two terms. Agrees with A145751 for all terms listed there (up to 65536).
Binomial transform is A078057 without initial 1, second binomial transform is A048580, third binomial transform is A163606, fourth binomial transform is A163604, fifth binomial transform is A163605.
a(n) is the number of vertices of the (n-1)-iterated line digraph L^{n-1}(G) of the digraph G(=L^0(G)) with vertices u,v,w and arcs u->v, v->u, v->w, w->v. - Miquel A. Fiol, Jun 08 2024

Links

Crossrefs

Cf. A000079 (powers of 2), A007283 (3*2^n), A027383, A029744, A048580, A052955, A063759, A078057, A090989, A145751, A163604, A163605, A163606.

Programs

  • Magma
    [ n le 2 select n+2 else 2*Self(n-2): n in [1..41] ];
  • Mathematica
    LinearRecurrence[{0,2}, {3,4}, 52] (* or *) Table[(1/2)*(5-(-1)^n )*2^((2*n-1+(-1)^n)/4), {n,50}] (* G. C. Greubel, Aug 24 2017 *)
  • PARI
    my(x='x+O('x^50)); Vec(x*(3+4*x)/(1-2*x^2)) \\ G. C. Greubel, Aug 24 2017
  • SageMath
    [(2+(n%2))*2^((n-(n%2))//2) for n in range(1,41)] # G. C. Greubel, Jun 13 2024

Formula

a(n) = A027383(n-1) + 2.
a(n) = A052955(n) + 1 for n >= 1.
a(n) = (1/2)*(5 - (-1)^n)*2^((2*n - 1 + (-1)^n)/4).
G.f.: x*(3+4*x)/(1-2*x^2).
a(n) = A090989(n-1).
E.g.f.: (1/2)*(4*cosh(sqrt(2)*x) + 3*sqrt(2)*sinh(sqrt(2)*x) - 4). - G. C. Greubel, Aug 24 2017
a(n) = A063759(n), n >= 1. - R. J. Mathar, Jan 25 2023

A164021 a(n) = 12*a(n-1) - 34*a(n-2) for n > 1; a(0) = 3, a(1) = 22.

Original entry on oeis.org

3, 22, 162, 1196, 8844, 65464, 484872, 3592688, 26626608, 197367904, 1463110176, 10846813376, 80416014528, 596200519552, 4420261740672, 32772323223296, 242978979496704, 1801488764368384, 13356579869532672, 99028340445867008
Offset: 0

Views

Author

Klaus Brockhaus, Aug 08 2009

Keywords

Comments

Binomial transform of A163605.

Links

Crossrefs

Cf. A163605.

Programs

  • Magma
    [ n le 2 select 19*n-16 else 12*Self(n-1)-34*Self(n-2): n in [1..20] ];
  • Mathematica
    LinearRecurrence[{12,-34},{3,22},20] (* Harvey P. Dale, Oct 19 2012 *)
  • PARI
    x='x+O('x^50); Vec((3-14*x)/(1-12*x+34*x^2)) \\ G. C. Greubel, Sep 07 2017

Formula

a(n) = ((3+2*sqrt(2))*(6+sqrt(2))^n+(3-2*sqrt(2))*(6-sqrt(2))^n)/2.
G.f.: (3-14*x)/(1-12*x+34*x^2).
E.g.f.: (3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x))*exp(6*x). - G. C. Greubel, Sep 07 2017
Showing 1-3 of 3 results.

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