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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.

%I A250111 #38 Sep 08 2022 08:46:10
%S A250111 1,1,1,3,4,9,13,25,38,68,106,182,288,483,771,1275,2046,3355,5401,8811,
%T A250111 14212,23112,37324,60580,97904,158717,256621,415715,672336,1088661,
%U A250111 1760997,2850645,4611642,7463884,12075526,19541994,31617520,51163695,82781215
%N A250111 Number of orbits of size 2 in vertices of Fibonacci cube Gamma_n under the action of its automorphism group.
%H A250111 Colin Barker, <a href="/A250111/b250111.txt">Table of n, a(n) for n = 1..1000</a>
%H A250111 A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-August-11-2014.pdf">Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings</a>, 2014.
%H A250111 A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-submit-2014.pdf">Vertex and edge orbits of Fibonacci and Lucas cubes</a>, 2014; See Table 1.
%H A250111 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,0,-1,-1).
%F A250111 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
%F A250111 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
%F A250111 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
%F A250111 From _G. C. Greubel_, Apr 06 2022: (Start)
%F A250111 a(n) = [n=1] + Sum_{k=0..floor((n-1)/2)} Fibonacci(k+1)*Fibonacci(n-2*k-1).
%F A250111 a(2*n) = (1/2)*(Fibonacci(2*n+2) - Fibonacci(n+1)), n >= 1.
%F A250111 a(2*n+1) = (1/2)*(Fibonacci(2*n+3) - Fibonacci(n+3) + 2*[n=0]), n >= 0. (End)
%t A250111 LinearRecurrence[{1,2,-1,0,-1,-1},{1,1,1,3,4,9,13},40] (* _Harvey P. Dale_, Feb 10 2018 *)
%o A250111 (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
%o A250111 (PARI) a(n)=if(n==1,1,(fibonacci(n+2) - fibonacci((n-(-1)^n)\2+2))/2); \\ _Joerg Arndt_, Nov 22 2014
%o A250111 (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
%o A250111 (SageMath)
%o A250111 def A250111(n): return bool(n==1) + sum( fibonacci(j+1)*fibonacci(n-2*j-1) for j in (0..((n-1)//2)) )
%o A250111 [A250111(n) for n in (1..50)] # _G. C. Greubel_, Apr 06 2022
%Y A250111 Cf. A000045, A058071.
%K A250111 nonn,easy
%O A250111 1,4
%A A250111 _N. J. A. Sloane_, Nov 19 2014
%E A250111 More terms from _Vincenzo Librandi_, Nov 22 2014