login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A099919 F(3) + F(6) + F(9) + ... + F(3n), F(n) = Fibonacci numbers A000045. 14
0, 2, 10, 44, 188, 798, 3382, 14328, 60696, 257114, 1089154, 4613732, 19544084, 82790070, 350704366, 1485607536, 6293134512, 26658145586, 112925716858, 478361013020, 2026369768940, 8583840088782, 36361730124070 (list; graph; refs; listen; history; text; internal format)
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.

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. 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

KEYWORD

nonn,easy

AUTHOR

Ralf Stephan, Oct 30 2004

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 15:09 EST 2019. Contains 329896 sequences. (Running on oeis4.)