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

1,4

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

Cf. A000045, A058071.

Sequence in context: A124285 A131326 A089300 * A079284 A000624 A244703

Adjacent sequences: A250108 A250109 A250110 * A250112 A250113 A250114

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 19 2014

Extensions

More terms from Vincenzo Librandi, Nov 22 2014

Status

approved