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%I #15 Sep 08 2022 08:46:16
%S 1,5,32,135,605,2560,10933,46305,196384,831875,3524489,14929920,
%T 63245753,267913165,1134902560,4807524015,20365009477,86267563520,
%U 365435291981,1548008735625,6557470308896,27777889982155,117669030432337,498454011740160,2111485077903025
%N a(n) = Fibonacci(3*n) - (2 + (-1)^n)*Fibonacci(n).
%C This is a divisibility sequence: if n divides m then a(n) divides a(m). The sequence satisfies a linear recurrence of order 6. In general, for integers r and s, the sequence Fibonacci(r*n) - 2*Fibonacci((r - 2*s)*n) + Fibonacci((r - 4*s)*n) is a divisibility sequence of the sixth order. This is the case r = 3, s = 1. See A127595 (case r = 4, s = 1).
%H G. C. Greubel, <a href="/A273623/b273623.txt">Table of n, a(n) for n = 1..1000</a>
%H P. Bala, <a href="/A273622/a273622.pdf">Lucas sequences and divisibility sequences</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-12,-4,4,1).
%F a(n) = Fibonacci(3*n) - 2*Fibonacci(n) + Fibonacci(-n).
%F a(2*n) = 5*Fibonacci(2*n)^3;
%F a(2n+1) = Fibonacci(2*n+1)*(5*Fibonacci(2*n+1)^2 - 4) = Fibonacci(2*n+1)*Lucas(2*n+1)^2.
%F O.g.f. x*(x^4 - x^3 + 8*x^2 + x + 1)/( (1 + x - x^2 )*(1 - x - x^2)*(1 - 4*x - x^2 ) ).
%F a(n) = 4*a(n-1) + 4*a(n-2) - 12*a(n-3) - 4*a(n-4) + 4*a(n-5) + a(n-6). - _G. C. Greubel_, Jun 02 2016
%p #A273623
%p with(combinat):
%p seq(fibonacci(3*n) - (2 + (-1)^n)*fibonacci(n), n = 1..25);
%t LinearRecurrence[{4, 4, -12, -4, 4, 1}, {1, 5, 32, 135, 605, 2560}, 100] (* _G. C. Greubel_, Jun 02 2016 *)
%t Table[Fibonacci[3 n] - (2 + (-1)^n) Fibonacci[n], {n, 1, 30}] (* _Vincenzo Librandi_, Jun 02 2016 *)
%o (Magma) [Fibonacci(3*n)-(2+(-1)^n)*Fibonacci(n): n in [1..25]]; // _Vincenzo Librandi_, Jun 02 2016
%o (PARI) a(n)=fibonacci(3*n) - (2 + (-1)^n)*fibonacci(n) \\ _Charles R Greathouse IV_, Jun 08 2016
%Y Cf. A000032, A000045, A127595, A273622, A273624, A273625.
%K nonn,easy
%O 1,2
%A _Peter Bala_, May 29 2016
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