A250111 Number of orbits of size 2 in vertices of Fibonacci cube Gamma_n under the action of its automorphism group.
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
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).
Programs
-
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
-
Mathematica
LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,1,3,4,9,13},40] (* Harvey P. Dale, Feb 10 2018 *) -
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
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)
Extensions
More terms from Vincenzo Librandi, Nov 22 2014