A273625 a(n) = (1/12)*(Fibonacci(2*n) + Fibonacci(4*n) + Fibonacci(6*n)).

1, 14, 228, 3948, 69905, 1248072, 22352707, 400808856, 7190208684, 129009258070, 2314882621811, 41538234954384, 745368939599413, 13375072472343218, 240005728531700340, 4306726622089196592, 77281063743045412517, 1386752354089549205976, 24884260852952644076119

Offset: 1

Author

Peter Bala, May 31 2016

Keywords

• nonn
• easy

Comments

This is a divisibility sequence, that is, if n divides m then a(n) divides a(m). The sequence satisfies a sixth-order linear recurrence. More generally, the sequence s(n) := Fibonacci(2*n) + Fibonacci(4*n) + ... + Fibonacci(2*k*n) is a divisibility sequence for k = 1,2,3,.... See A215466 for the case k = 2. Cf. A273623, A273624.

From Peter Bala, Aug 05 2019: (Start)

Let U(n;P,Q), where P and Q are integer parameters, denote the Lucas sequence of the first kind. Then, excluding the cases P = -1 and P = 0, the sequence ( U(n;P,1) + U(2*n;P,1) + U(3*n;P,1))/(P^2 + P) is a sixth-order linear divisibility sequence with o.g.f. x*(1 - 2*(P^2 - 2)*x + (3*P^3 - 3*P^2 - 8*P + 10)*x^2 - 2*(P^2 - 2)*x^3 + x^4)/((1 - P*x + x^2)*(1 - (P^2 - 2)*x + x^2)*(1 - P*(P^2 - 3)*x + x^2)). This is the case P = 3.

More generally, the sequence U(n;P,1) + U(2*n;P,1) + ... + U(k*n;P,1) is a linear divisibility sequence of order 2*k. See, for example, A215466 with P = 3, k = 2. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 1..795
• P. Bala, Lucas sequences and divisibility sequences
• P. Bala, Divisibility sequences from strong divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (28,-204,434,-204,28,-1).

Crossrefs

Cf. A000045, A001906, A215466, A273623, A273624, A215466.

Programs

Magma

[1/12*(Fibonacci(2*n)+Fibonacci(4*n)+Fibonacci(6*n)): n in [1..25]]; // Vincenzo Librandi, Jun 02 2016

Maple

#A273625
with(combinat):
seq(1/12*(fibonacci(2*n) + fibonacci(4*n) + fibonacci(6*n)), n = 1..20);

Mathematica

LinearRecurrence[{28, -204, 434, -204, 28, -1},{1, 14, 228, 3948, 69905, 1248072}, 100] (* G. C. Greubel, Jun 02 2016 *)
Table[1/12 (Fibonacci[2 n] + Fibonacci[4 n] + Fibonacci[6 n]), {n, 1, 30}] (* Vincenzo Librandi, Jun 02 2016 *)

PARI

A001906(n)=fibonacci(2*n)
a(n)=(A001906(n)+A001906(2*n)+A001906(3*n))/12 \\ Charles R Greathouse IV, Jun 08 2016

Formula

a(n) = -a(-n).

O.g.f.: x*(x^4 - 14*x^3 + 40*x^2 - 14*x + 1)/((x^2 - 3*x + 1)*(x^2 - 7*x + 1)*(x^2 - 18*x + 1)).

a(n) = 28*a(n-1) - 204*a(n-2) + 434*a(n-3) - 204*a(n-4) + 28*a(n-5) - a(n-6). - G. C. Greubel, Jun 02 2016