A103433 a(n) = Sum_{i=1..n} Fibonacci(2i-1)^2.

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

Offset: 0

Author

Ralf Stephan, Feb 08 2005

Keywords

• nonn
• easy

References

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

Links

• 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).

Crossrefs

Partial sums of A081068. Bisection of A077916.

Programs

Magma

[(1/5)*(Fibonacci(4*n)+2*n): n in [0..50]]; // Vincenzo Librandi, Apr 20 2011

Mathematica

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

PARI

a(n)=(fibonacci(4*n)+2*n)/5 \\ Charles R Greathouse IV, Oct 07 2015

Formula

G.f.: x*(1-4*x+x^2) / ((1-7*x+x^2)(1-x)^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..2n-1} (-1)^i*F(i)*F(i+1) |, where F(n) = Fibonacci numbers (A000045). - Rigoberto Florez, May 04 2019