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A099919
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F(3) + F(6) + F(9) + ... + F(3n), F(n) = Fibonacci numbers A000045.
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14
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0, 2, 10, 44, 188, 798, 3382, 14328, 60696, 257114, 1089154, 4613732, 19544084, 82790070, 350704366, 1485607536, 6293134512, 26658145586, 112925716858, 478361013020, 2026369768940, 8583840088782, 36361730124070
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OFFSET
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0,2
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COMMENTS
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Partial sum of the even Fibonacci numbers. - Vladimir Joseph Stephan Orlovsky, Nov 28 2010
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REFERENCES
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A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 25.
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Project Euler, Problem 2.
Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).
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FORMULA
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a(n) = (Fibonacci(3*n + 2) - 1)/2 = (A015448(n+1)-1)/2.
G.f.: 2*x/((1 - x)*(1 - 4*x - x^2)).
a(n) = (F(3n + 2) - 1)/2 = 2 * A049652(n).
a(n) = Sum_{0 <= j <= i <= n} binomial(i, j)*F(i + j). - Benoit Cloitre, May 21 2005
a(n) = 4*a(n - 1) + a(n - 2) + 2, n > 1. - Gary Detlefs, Dec 08 2010
a(n) = 5*a(n - 1) - 3*a(n - 2) - a(n - 3), n > 2. - Gary Detlefs, Dec 08 2010
a(n) = (Fibonacci(3*n + 3) + Fibonacci(3*n) - 2)/4. - Gary Detlefs, Dec 08 2010
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
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MATHEMATICA
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CoefficientList[Series[2 x/((1 - x) (1 - 4 x - x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)
LinearRecurrence[{5, -3, -1}, {0, 2, 10}, 30] (* G. C. Greubel, Jan 17 2018 *)
Accumulate[Fibonacci[3Range[0, 19]]] (* Alonso del Arte, Dec 23 2018 *)
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PROG
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(PARI) a(n) = sum(i=1, n, fibonacci(3*i)); \\ Michel Marcus, Mar 15 2014
(PARI) a(n) = fibonacci(3*n+2)\2 \\ Charles R Greathouse IV, Jun 11 2015
(MAGMA) [(Fibonacci(3*n+2) - 1)/2: n in [0..30]]; // G. C. Greubel, Jan 17 2018
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CROSSREFS
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Partial sums of A014445. Cf. A004794.
Cf. A087635.
Case k = 3 of partial sums of fibonacci(k*n): A000071, A027941, A058038, A138134, A053606.
Sequence in context: A243965 A218780 A068551 * A100397 A084059 A084609
Adjacent sequences: A099916 A099917 A099918 * A099920 A099921 A099922
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KEYWORD
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nonn,easy
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AUTHOR
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Ralf Stephan, Oct 30 2004
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EXTENSIONS
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a(0) = 0 prepended by Joerg Arndt, Mar 13 2014
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STATUS
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approved
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