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A273625 a(n) = (1/12)*(Fibonacci(2*n) + Fibonacci(4*n) + Fibonacci(6*n)). 4
1, 14, 228, 3948, 69905, 1248072, 22352707, 400808856, 7190208684, 129009258070, 2314882621811, 41538234954384, 745368939599413, 13375072472343218, 240005728531700340, 4306726622089196592, 77281063743045412517, 1386752354089549205976, 24884260852952644076119 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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).

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

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 *)

PROG

(MAGMA) [1/12*(Fibonacci(2*n)+Fibonacci(4*n)+Fibonacci(6*n)): n in [1..25]]; // 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

CROSSREFS

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

Sequence in context: A320762 A027774 A099272 * A120048 A079563 A230346

Adjacent sequences:  A273622 A273623 A273624 * A273626 A273627 A273628

KEYWORD

nonn,easy

AUTHOR

Peter Bala, May 31 2016

STATUS

approved

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Last modified October 21 11:42 EDT 2019. Contains 328296 sequences. (Running on oeis4.)