

A103433


a(n) = Sum_{i=1..n} Fibonacci(2i1)^2.


6



0, 1, 5, 30, 199, 1355, 9276, 63565, 435665, 2986074, 20466835, 140281751, 961505400, 6590256025, 45170286749, 309601751190, 2122041971551, 14544692049635, 99690802375860, 683290924581349, 4683345669693545
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OFFSET

0,3


REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 234.


LINKS

Table of n, a(n) for n=0..20.
Belgacem Bouras, A New Characterization of Catalan Numbers Related to Hankel Transforms and Fibonacci Numbers, Journal of Integer Sequences, 16 (2013), #13.3.3.
M. Dougherty, C. French, B. Saderholm, W. Qian,, Hankel Transforms of Linear Combinations of Catalan Numbers, J. Int. Seq. 14 (2011) # 11.5.1.
Index entries for linear recurrences with constant coefficients, signature (9,16,9,1).


FORMULA

G.f.: x*(14*x+x^2) / ((17*x+x^2)(1x)^2).
a(n) = (1/5)*(Fibonacci(4n) + 2n).
a(n) = (floor(5*n*phi) + 4*Fibonacci(4*n))/20, where phi =(1+sqrt(5))/2.  Gary Detlefs, Mar 10 2011
a(n) = (8*(n+2)*(Sum_{k=1..n} 1/(2*k^2 + 6*k + 4)) + Fibonacci(4*n))/5.  Gary Detlefs, Dec 07 2011
a(n) =  Sum_{i=0..2n1} (1)^i*F(i)*F(i+1) , where F(n) = Fibonacci numbers (A000045).  Rigoberto Florez, May 04 2019


MATHEMATICA

Table[(Fibonacci[4n]+2n)/5, {n, 0, 20}] (* Rigoberto Florez, May 04 2019 *)


PROG

(MAGMA) [(1/5)*(Fibonacci(4*n)+2*n): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011
(PARI) a(n)=(fibonacci(4*n)+2*n)/5 \\ Charles R Greathouse IV, Oct 07 2015


CROSSREFS

Partial sums of A081068. Bisection of A077916.
Sequence in context: A265279 A034164 A322257 * A081015 A090139 A107265
Adjacent sequences: A103430 A103431 A103432 * A103434 A103435 A103436


KEYWORD

nonn,easy


AUTHOR

Ralf Stephan, Feb 08 2005


STATUS

approved



