Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #68 Jun 03 2024 16:07:32
%S 0,2,10,44,188,798,3382,14328,60696,257114,1089154,4613732,19544084,
%T 82790070,350704366,1485607536,6293134512,26658145586,112925716858,
%U 478361013020,2026369768940,8583840088782,36361730124070,154030760585064,652484772464328,2763969850442378
%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 From _Gary Detlefs_, Dec 08 2010: (Start)
%F a(n) = 4*a(n - 1) + a(n - 2) + 2, n > 1.
%F a(n) = 5*a(n - 1) - 3*a(n - 2) - a(n - 3), n > 2.
%F a(n) = (Fibonacci(3*n + 3) + Fibonacci(3*n) - 2)/4. (End)
%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
%F E.g.f.: exp(x)*(exp(x)*(5*cosh(sqrt(5)*x) + 3*sqrt(5)*sinh(sqrt(5)*x)) - 5)/10. - _Stefano Spezia_, Jun 03 2024
%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, A015448, A049652.
%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