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%I #23 Sep 08 2022 08:46:16
%S 1,14,228,3948,69905,1248072,22352707,400808856,7190208684,
%T 129009258070,2314882621811,41538234954384,745368939599413,
%U 13375072472343218,240005728531700340,4306726622089196592,77281063743045412517,1386752354089549205976,24884260852952644076119
%N a(n) = (1/12)*(Fibonacci(2*n) + Fibonacci(4*n) + Fibonacci(6*n)).
%C 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.
%C From _Peter Bala_, Aug 05 2019: (Start)
%C 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.
%C 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)
%H G. C. Greubel, <a href="/A273625/b273625.txt">Table of n, a(n) for n = 1..795</a>
%H P. Bala, <a href="/A273622/a273622.pdf">Lucas sequences and divisibility sequences</a>
%H P. Bala, <a href="/A238600/a238600_1.pdf">Divisibility sequences from strong divisibility sequences</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (28,-204,434,-204,28,-1).
%F a(n) = -a(-n).
%F 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)).
%F 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
%p #A273625
%p with(combinat):
%p seq(1/12*(fibonacci(2*n) + fibonacci(4*n) + fibonacci(6*n)), n = 1..20);
%t LinearRecurrence[{28, -204, 434, -204, 28, -1},{1, 14, 228, 3948, 69905, 1248072}, 100] (* _G. C. Greubel_, Jun 02 2016 *)
%t Table[1/12 (Fibonacci[2 n] + Fibonacci[4 n] + Fibonacci[6 n]), {n, 1, 30}] (* _Vincenzo Librandi_, Jun 02 2016 *)
%o (Magma) [1/12*(Fibonacci(2*n)+Fibonacci(4*n)+Fibonacci(6*n)): n in [1..25]]; // _Vincenzo Librandi_, Jun 02 2016
%o (PARI) A001906(n)=fibonacci(2*n)
%o a(n)=(A001906(n)+A001906(2*n)+A001906(3*n))/12 \\ _Charles R Greathouse IV_, Jun 08 2016
%Y Cf. A000045, A001906, A215466, A273623, A273624, A215466.
%K nonn,easy
%O 1,2
%A _Peter Bala_, May 31 2016
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