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.

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A128052 a(n) = (F(2*n-1) + F(2*n+1))*(5/6 - cos(2*Pi*n/3)/3), where F(n) = Fibonacci(n).

Original entry on oeis.org

1, 3, 7, 9, 47, 123, 161, 843, 2207, 2889, 15127, 39603, 51841, 271443, 710647, 930249, 4870847, 12752043, 16692641, 87403803, 228826127, 299537289, 1568397607, 4106118243, 5374978561, 28143753123, 73681302247, 96450076809, 505019158607, 1322157322203
Offset: 0

Views

Author

Paul Barry, Feb 13 2007

Keywords

Comments

The a(n+1) are the numerators of A178381(4*n+3)/A178381(4*n+2). For the denominators see A179133(n). - Johannes W. Meijer, Jul 01 2010

Links

Crossrefs

Cf. A128053.
Cf. A179134. Trisection: A023039.

Programs

  • Magma
    I:=[1,3,7,9,47,123]; [n le 6 select I[n] else 18*Self(n-3)-Self(n-6): n in [1..30]]; // Vincenzo Librandi, Jul 17 2019
  • Maple
    with(combinat): nmax:=25; for n from 0 to nmax do a(n):= (fibonacci(2*n-1)+fibonacci(2*n+1))*(5/6-cos(2*Pi*n/3)/3) od: seq(a(n),n=0..nmax); # Johannes W. Meijer, Jul 01 2010
  • Mathematica
    LinearRecurrence[{0, 0, 18, 0, 0, -1}, {1, 3, 7, 9, 47, 123}, 40] (* Vincenzo Librandi, Jul 17 2019 *)

Formula

Lim_{n->infinity} A128052(n+1)/A179133(n) = 1 + cos(Pi/5). - Johannes W. Meijer, Jul 01 2010
a(n) = Lucas(2*n)*(Fibonacci(n) mod 2 + 1)/2, Lucas(n)=A000032, Fibonacci(n)=A000045. - Gary Detlefs, Jan 19 2001
From Colin Barker, Jun 27 2013: (Start)
a(n) = 18*a(n-3) - a(n-6).
G.f: -(3*x^5 + 7*x^4 + 9*x^3 - 7*x^2 - 3*x - 1) / ((x^2 - 3*x + 1)*(x^4 + 3*x^3 + 8*x^2 + 3*x + 1)). (End)
With L(n) the Lucas number A000032, a(n) = L(2*n)/2 or L(2*n) according as n is, or is not, divisible by 3. - David Callan, Jul 17 2019

A179133 Denominators of A178381(4*n+3)/A178381(4*n+2).

Original entry on oeis.org

2, 4, 5, 26, 68, 89, 466, 1220, 1597, 8362, 21892, 28657, 150050, 392836, 514229, 2692538, 7049156, 9227465, 48315634, 126491972, 165580141, 866988874, 2269806340, 2971215073, 15557484098, 40730022148, 53316291173, 279167724890
Offset: 0

Views

Author

Johannes W. Meijer, Jul 01 2010

Keywords

Comments

For the numerators see A128052.

Links

Crossrefs

Cf. A000045, A128052, A153727, A178381, A179131, A179132, A179133, A179134, A296182.

Programs

  • Maple
    with(GraphTheory): nmax:=120; P:=9: G:=PathGraph(P): A:= AdjacencyMatrix(G): for n from 0 to nmax do B(n):=A^n; A178381(n):=add(B(n)[1,k],k=1..P); od: for n from 0 to nmax-1 do a(n):= denom(A178381(4*n+3)/A178381(4*n+2)) od: seq(a(n),n=0..nmax/4-1);
  • Mathematica
    Flatten[Table[{2*Fibonacci[6 n + 1], 2*Fibonacci[6 n + 3],
Fibonacci[6 n + 5]}, {n, 0, 10}]] (* Greg Dresden, Oct 16 2021 *)
LinearRecurrence[{0,0,18,0,0,-1},{2,4,5,26,68,89},30] (* Harvey P. Dale, Oct 08 2024 *)

Formula

a(n) = A179134(n)*A153727(n+1)/2.
Lim_{n->infinity} A128052(n+1)/A179133(n) = 1+cos(Pi/5) = A296182.
From Colin Barker, Jun 27 2013: (Start)
G.f.: -(x^5+4*x^4+10*x^3-5*x^2-4*x-2)/((x^2-3*x+1)*(x^4+3*x^3+8*x^2+3*x+1)).
a(n) = 18*a(n-3)-a(n-6). (End)
From Greg Dresden, Oct 16 2021: (Start)
a(3*n) = 2*Fibonacci(6*n+1),
a(3*n+1) = 2*Fibonacci(6*n+3),
a(3*n+2) = Fibonacci(6*n+5). (End)
Showing 1-2 of 2 results.

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