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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A099015 a(n) = Fib(n+1)*(2*Fib(n)^2 + Fib(n)*Fib(n-1) + Fib(n-1)^2).

Original entry on oeis.org

1, 2, 8, 33, 140, 592, 2509, 10626, 45016, 190685, 807764, 3421728, 14494697, 61400482, 260096680, 1101787113, 4667245276, 19770767984, 83750317589, 354772037730, 1502838469496, 6366125914117, 26967342128548
Offset: 0

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Author

Paul Barry, Sep 22 2004

Keywords

Comments

Form the matrix A=[1,1,1,1;3,2,1,0;3,1,0,0;1,0,0,0]=(binomial(3-j,i)). Then a(n)=(2,2)-element of A^n.

Links

Crossrefs

Cf. A000045, A056570, A066258, A066259, A099014, A033887.

Programs

  • Magma
    [Fibonacci(n+1)*(2*Fibonacci(n)^2 + Fibonacci(n)*Fibonacci(n-1) + Fibonacci(n-1)^2): n in [0..30]]; // Vincenzo Librandi, Jun 05 2011
  • Mathematica
    LinearRecurrence[{3,6,-3,-1},{1,2,8,33},30] (* Harvey P. Dale, Nov 28 2015 *)
CoefficientList[Series[(1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Dec 31 2017 *)
  • PARI
    a(n)=my(e=fibonacci(n-1),f=fibonacci(n));(e+f)*(2*f^2+f*e+e^2) \\ Charles R Greathouse IV, Jun 05 2011
  • PARI
    first(n) = Vec((1 - x - 4*x^2)/(1 - 3*x - 6*x^2 + 3*x^3 + x^4) + O(x^n)) \\ Iain Fox, Dec 31 2017

Formula

G.f.: (1-x-4*x^2)/((1+x-x^2)*(1-4*x-x^2)).
G.f.: (1-x-4*x^2)/(1-3*x-6*x^2+3*x^3+x^4).
a(n) = (3*Fib(3*n+1) + (-1)^n*Fib(n-3))/5.
a(n) = (2+sqrt(5))^n*(3/10 + 3*sqrt(5)/50) + (2-sqrt(5))^n*(3/10 - 3*sqrt(5)/50) + (-1)^n*((1/2 - sqrt(5)/2)^n*(1/5 + 2*sqrt(5)/25) + (1/5 - 2*sqrt(5)/25)*(1/2 + sqrt(5)/2)^n).

A358917 a(n) = Fibonacci(n+1)^4 - Fibonacci(n-1)^4.

Original entry on oeis.org

0, 1, 15, 80, 609, 4015, 27936, 190385, 1307775, 8956144, 61405905, 420831071, 2884553280, 19770670945, 135511114479, 928804587920, 6366127657281, 43634071586575, 299072419071840, 2049872742473489, 14050037090947935, 96300386075488816, 660052667580788145
Offset: 0

Views

Author

Feryal Alayont, Dec 05 2022

Keywords

Comments

a(n) is the number of edge covers of a spider graph with four branches where each branch has n vertices besides the center vertex.
An edge cover of a graph is a subset of edges for which each vertex is incident to at least one edge in the subset.
The idea is each branch is treated as a path P_(n+2). Each branch acts independently then and has F(n+1) covers (P_n has F(n-1) covers), hence F(n+1)^4 total. Except we remove the cases where each branch is missing the connecting edge to the center, which is when that edge cover comes from P_n , hence the minus F(n-1)^4.

Examples

			Case n=1 is a star graph with four branches and one edge cover (all edges).
   *   *
    \ /
  *__C__*
For n=2, there are 15 edge covers of the graph obtained by gluing four P_3 paths at one single vertex. Each of the pendant edges of the P_3's have to be in the edge cover for the pendants to be incident with an edge. The middle vertices are then automatically incident with at least one edge. There remains the center vertex. We then need at least one of the remaining four edges to be in the subset, giving us 2^4-1 choices.
    *       *
     \     /
      *   *
       \ /
  *__*__C__*__*

Links

Crossrefs

Cf. A000032, A000045, A056571, A099014, A358934.

Programs

  • Mathematica
    LinearRecurrence[{4, 19, 4, -1}, {0, 1, 15, 80}, 25] (* Amiram Eldar, Dec 06 2022 *)
  • Python
    from sympy import fibonacci
def a(n): return fibonacci(n+1)**4-fibonacci(n-1)**4

Formula

From Stefano Spezia, Dec 06 2022: (Start)
G.f.: x*(1 + 11*x + x^2)/((1 - 7*x + x^2)*(1 + 3*x + x^2)).
a(n) = 4*a(n-1) + 19*a(n-2) + 4*a(n-3) - a(n-4) for n > 3. (End)
5*a(n) = 9*A004187(n) + 4*(-1)^n*A001906(n) . - R. J. Mathar, May 07 2024
a(n) = L(n) * A099014(n). - Greg Dresden and Chuanqi Wei, Sep 04 2025
Showing 1-2 of 2 results.

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