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%I #62 Sep 28 2023 08:18:36
%S 0,2,10,44,188,798,3382,14328,60696,257114,1089154,4613732,19544084,
%T 82790070,350704366,1485607536,6293134512,26658145586,112925716858,
%U 478361013020,2026369768940,8583840088782,36361730124070
%N a(n) = F(3) + F(6) + F(9) + ... + F(3n), F(n) = Fibonacci numbers A000045.
%C Partial sum of the even Fibonacci numbers. - _Vladimir Joseph Stephan Orlovsky_, Nov 28 2010
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 25.
%H Vincenzo Librandi, <a href="/A099919/b099919.txt">Table of n, a(n) for n = 0..1000</a>
%H Project Euler, <a href="https://projecteuler.net/problem=2">Problem 2: Even Fibonacci Numbers</a>.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-1).
%F a(n) = (Fibonacci(3*n + 2) - 1)/2 = (A015448(n+1)-1)/2.
%F G.f.: 2*x/((1 - x)*(1 - 4*x - x^2)).
%F a(n) = (F(3n + 2) - 1)/2 = 2 * A049652(n).
%F a(n) = Sum_{0 <= j <= i <= n} binomial(i, j)*F(i + j). - _Benoit Cloitre_, May 21 2005
%F a(n) = 4*a(n - 1) + a(n - 2) + 2, n > 1. - _Gary Detlefs_, Dec 08 2010
%F a(n) = 5*a(n - 1) - 3*a(n - 2) - a(n - 3), n > 2. - _Gary Detlefs_, Dec 08 2010
%F a(n) = (Fibonacci(3*n + 3) + Fibonacci(3*n) - 2)/4. - _Gary Detlefs_, Dec 08 2010
%F a(n) = (-10 + (5 - 3*sqrt(5))*(2 - sqrt(5))^n + (2 + sqrt(5))^n*(5 + 3*sqrt(5)))/20. - _Colin Barker_, Nov 26 2016
%t CoefficientList[Series[2 x/((1 - x) (1 - 4 x - x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 15 2014 *)
%t LinearRecurrence[{5, -3, -1}, {0, 2, 10}, 30] (* _G. C. Greubel_, Jan 17 2018 *)
%t Accumulate[Fibonacci[3Range[0, 19]]] (* _Alonso del Arte_, Dec 23 2018 *)
%o (PARI) a(n) = sum(i=1, n, fibonacci(3*i)); \\ _Michel Marcus_, Mar 15 2014
%o (PARI) a(n) = fibonacci(3*n+2)\2 \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma) [(Fibonacci(3*n+2) - 1)/2: n in [0..30]]; // _G. C. Greubel_, Jan 17 2018
%Y Partial sums of A014445.
%Y Cf. A000045, A004794.
%Y Cf. A087635.
%Y Case k = 3 of partial sums of fibonacci(k*n): A000071, A027941, A058038, A138134, A053606.
%K nonn,easy
%O 0,2
%A _Ralf Stephan_, Oct 30 2004
%E a(0) = 0 prepended by _Joerg Arndt_, Mar 13 2014
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