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A250111
Number of orbits of size 2 in vertices of Fibonacci cube Gamma_n under the action of its automorphism group.
2
1, 1, 1, 3, 4, 9, 13, 25, 38, 68, 106, 182, 288, 483, 771, 1275, 2046, 3355, 5401, 8811, 14212, 23112, 37324, 60580, 97904, 158717, 256621, 415715, 672336, 1088661, 1760997, 2850645, 4611642, 7463884, 12075526, 19541994, 31617520, 51163695, 82781215
OFFSET
1,4
LINKS
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.
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
Sequence in context: A124285 A131326 A089300 * A079284 A000624 A244703
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 19 2014
EXTENSIONS
More terms from Vincenzo Librandi, Nov 22 2014
STATUS
approved