OFFSET
1,4
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 1.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,0,-1,-1).
FORMULA
a(n) = (1/2) * (F(n+2) - F(floor((n-(-1)^n)/2)+2)) for n >= 2, a(1)=1. - Joerg Arndt, Nov 22 2014
a(n) = a(n-1)+2*a(n-2)-a(n-3)-a(n-5)-a(n-6) for n>7. - Colin Barker, Dec 01 2014
G.f.: x*(1-2*x^2+x^3+x^5+x^6)/((1-x-x^2)*(1-x^2-x^4)). - Colin Barker, Dec 01 2014
From G. C. Greubel, Apr 06 2022: (Start)
a(n) = [n=1] + Sum_{k=0..floor((n-1)/2)} Fibonacci(k+1)*Fibonacci(n-2*k-1).
a(2*n) = (1/2)*(Fibonacci(2*n+2) - Fibonacci(n+1)), n >= 1.
a(2*n+1) = (1/2)*(Fibonacci(2*n+3) - Fibonacci(n+3) + 2*[n=0]), n >= 0. (End)
MATHEMATICA
LinearRecurrence[{1, 2, -1, 0, -1, -1}, {1, 1, 1, 3, 4, 9, 13}, 40] (* Harvey P. Dale, Feb 10 2018 *)
PROG
(Magma) [n eq 1 select 1 else (1/2)*(Fibonacci(n+2)-Fibonacci(Floor((n-(-1)^n)/2)+2)): n in [1..40]]; // Vincenzo Librandi, Nov 22 2014
(PARI) a(n)=if(n==1, 1, (fibonacci(n+2) - fibonacci((n-(-1)^n)\2+2))/2); \\ Joerg Arndt, Nov 22 2014
(PARI) Vec(x*(1-2*x^2+x^3+x^5+x^6)/((1-x-x^2)*(1-x^2-x^4)) + O(x^100)) \\ Colin Barker, Dec 01 2014
(SageMath)
def A250111(n): return bool(n==1) + sum( fibonacci(j+1)*fibonacci(n-2*j-1) for j in (0..((n-1)//2)) )
[A250111(n) for n in (1..50)] # G. C. Greubel, Apr 06 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 19 2014
EXTENSIONS
More terms from Vincenzo Librandi, Nov 22 2014
STATUS
approved