A318404 a(n) = F(n+1)^4 - 4*F(n-1)*F(n)^3, where F(n) = A000045(n), the n-th Fibonacci number.

1, 1, 12, 49, 409, 2596, 18321, 124177, 854764, 5849089, 40115241, 274888516, 1884285217, 12914634529, 88519396044, 606717892561, 4158514347961, 28502860300132, 195361565985969, 1339027949145649, 9177834477168556, 62905812346085281, 431162854681140297

Offset

0,3

Comments

a(n) is the number of Markov equivalence classes whose skeleton is a spider graph with four legs, each of which contains n nodes of degree at most two.

A001519 admits the related formula A001519(n) = F(n+1)^2 - 2*F(n-1)*F(n).

Links

Robert Israel, Table of n, a(n) for n = 0..1195

A. Radhakrishnan, L. Solus, and C. Uhler. Counting Markov equivalence classes for DAG models on trees, arXiv:1706.06091 [math.CO], 2017; Discrete Applied Mathematics 244 (2018): 170-185.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Formula

G.f.: (-1 + 4*x + 8*x^2 + 11*x^3 - 4*x^4)/(-1 + 5*x + 15*x^2 - 15*x^3 - 5*x^4 + x^5). - Robert Israel, Aug 26 2018

Maple

f:= gfun:-rectoproc({a(n+5)-5*a(n+4)-15*a(n+3)+15*a(n+2)+5*a(n+1)-a(n), a(0)=1, a(1)=1, a(2)=12, a(3)=49, a(4)=409}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Aug 26 2018

Mathematica

Table[Fibonacci[n + 1]^4 - 4 Fibonacci[n - 1] Fibonacci[n]^3, {n, 0, 25}] (* Vincenzo Librandi, Aug 26 2018 *)
CoefficientList[Series[(-1 + 4 x + 8 x^2 + 11 x^3 - 4 x^4)/(-1 + 5 x + 15 x^2 - 15 x^3 - 5 x^4 + x^5), {x, 0, 50}], x] (* Stefano Spezia, Sep 03 2018 *)

Prog

(SageMath)
def a(n):
    return fibonacci(n+1)^4-4*fibonacci(n-1)*fibonacci(n)^3
[a(n) for n in range(20)]
(Magma) [Fibonacci(n+1)^4-4*Fibonacci(n-1)*Fibonacci(n)^3: n in [0..25]]; // Vincenzo Librandi, Aug 26 2018
(PARI) a(n) = fibonacci(n+1)^4 - 4*fibonacci(n-1)*fibonacci(n)^3; \\ Michel Marcus, Aug 26 2018

Crossrefs

Cf. A000045, A001519, A318376.

Sequence in context: A307921 A288516 A219153 * A041274 A029586 A335698

Adjacent sequences: A318401 A318402 A318403 * A318405 A318406 A318407

Keyword

nonn,easy

Author

Liam Solus, Aug 26 2018

Extensions

a(0) = 1 prepended by Vincenzo Librandi, Aug 26 2018

Status

approved