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

Author

Liam Solus, Aug 26 2018

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000045, A001519, A318376.

Programs

Magma

[Fibonacci(n+1)^4-4*Fibonacci(n-1)*Fibonacci(n)^3: n in [0..25]]; // Vincenzo Librandi, 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 *)

PARI

a(n) = fibonacci(n+1)^4 - 4*fibonacci(n-1)*fibonacci(n)^3; \\ Michel Marcus, Aug 26 2018

SageMath

def a(n):
    return fibonacci(n+1)^4-4*fibonacci(n-1)*fibonacci(n)^3
[a(n) for n in range(20)]

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

Extensions

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