A318376 -id:A318376 - cp's OEIS frontend

A318404 a(n) = F(n+1)^4 - 4F(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

Liam Solus, Aug 26 2018

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).

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).

Cf. A000045, A001519, A318376.

[Fibonacci(n+1)^4-4*Fibonacci(n-1)*Fibonacci(n)^3: n in [0..25]]; // Vincenzo Librandi, Aug 26 2018

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

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 *)

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

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

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

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

A318405 Rectangular array R read by antidiagonals: R(n,k) = F(n+1)^k - kF(n-1)F(n)^(k-1), where F(n) = A000045(n), the n-th Fibonacci number; n >= 0, k >= 1.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 5, 5, 3, 1, 1, 12, 15, 13, 5, 1, 1, 27, 49, 71, 34, 8, 1, 1, 58, 163, 409, 287, 89, 13, 1, 1, 121, 537, 2315, 2596, 1237, 233, 21, 1, 1, 248, 1739, 12709, 23393, 18321, 5205, 610, 34, 1, 1, 503, 5537, 67919, 205894, 268893, 124177, 22105, 1597, 55

Offset: 0

Liam Solus, Aug 26 2018

Row index n begins with 0, column index begins with 1.

R(n,k) is the number of Markov equivalence classes whose skeleton is a spider graph with k legs, each of which contains n nodes of degree at most two. See Corollary 4.2 in the paper by A. Radhakrishnan et al. below.

			The rectangular array R(n,k) begins:
n\k|   1      2      3        4         5          6            7
---+-------------------------------------------------------------
0  |   0      1      1        1         1          1            1
1  |   1      1      1        1         1          1            1
2  |   1      2      5       12        27         58          121
3  |   2      5     15       49       163        537         1739
4  |   3     13     71      409      2315      12709        67919
5  |   5     34    287     2596     23393     205894      1769027
6  |   8     89   1237    18321    268893    3843769     53573477
7  |  13    233   5205   124177   2941661   67944057   1530787237

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.

Columns include A000045, A001519, A318376, A318404.

Cf. A007984.

def R(n, k):
    return fibonacci(n+1)^k-k*fibonacci(n-1)*fibonacci(n)^(k-1)

cp's OEIS Frontend

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

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A318405 Rectangular array R read by antidiagonals: R(n,k) = F(n+1)^k - kF(n-1)F(n)^(k-1), where F(n) = A000045(n), the n-th Fibonacci number; n >= 0, k >= 1.

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Sage

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A318405 Rectangular array R read by antidiagonals: R(n,k) = F(n+1)^k - k*F(n-1)*F(n)^(k-1), where F(n) = A000045(n), the n-th Fibonacci number; n >= 0, k >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Sage

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

A318405 Rectangular array R read by antidiagonals: R(n,k) = F(n+1)^k - kF(n-1)F(n)^(k-1), where F(n) = A000045(n), the n-th Fibonacci number; n >= 0, k >= 1.