A099919 a(n) = F(3) + F(6) + F(9) + ... + F(3n), F(n) = Fibonacci numbers A000045.

0, 2, 10, 44, 188, 798, 3382, 14328, 60696, 257114, 1089154, 4613732, 19544084, 82790070, 350704366, 1485607536, 6293134512, 26658145586, 112925716858, 478361013020, 2026369768940, 8583840088782, 36361730124070

Offset

0,2

Comments

Partial sum of the even Fibonacci numbers. - Vladimir Joseph Stephan Orlovsky, Nov 28 2010

References

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 25.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Project Euler, Problem 2: Even Fibonacci Numbers.

Index entries for linear recurrences with constant coefficients, signature (5,-3,-1).

Formula

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

Mathematica

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 *)

Prog

(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

Crossrefs

Partial sums of A014445.

Cf. A000045, 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 A339642

Adjacent sequences: A099916 A099917 A099918 * A099920 A099921 A099922

Keyword

nonn,easy

Author

Ralf Stephan, Oct 30 2004

Extensions

a(0) = 0 prepended by Joerg Arndt, Mar 13 2014

Status

approved