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A318404 a(n) = F(n+1)^4 - 4*F(n-1)*F(n)^3, where F(n) = A000045(n), the n-th Fibonacci number. 2
1, 1, 12, 49, 409, 2596, 18321, 124177, 854764, 5849089, 40115241, 274888516, 1884285217, 12914634529, 88519396044, 606717892561, 4158514347961, 28502860300132, 195361565985969, 1339027949145649, 9177834477168556, 62905812346085281, 431162854681140297 (list; graph; refs; listen; history; text; internal format)
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

(SAGE)

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 A081292

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

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Last modified May 27 21:52 EDT 2020. Contains 334671 sequences. (Running on oeis4.)