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%I #16 Jun 13 2015 00:55:16
%S 0,0,5,7,22,37,81,143,276,490,895,1578,2802,4894,8547,14797,25560,
%T 43919,75267,128525,218930,371920,630465,1066452,1800612,3034812,
%U 5106881,8580883,14398426,24129145,40388085,67527563,112786512
%N Sum of the eccentricities of all vertices in the Lucas cube Lambda(n).
%C The vertex set of the Lucas cube Lambda(n) is the set of all binary strings of length n without consecutive 1's and without a 1 in the first and the last bit. Two vertices of the Lucas cube are adjacent if their strings differ in exactly one bit.
%C a(n) = Sum(k*A210572(n,k), k=0..n).
%H A. Castro and M. Mollard, <a href="http://dx.doi.org/10.1016/j.disc.2011.11.006">The eccentricity sequences of Fibonacci and Lucas cubes</a>, Discrete Math., 312 (2012), 1025-1037.
%H S. Klavzar, M. Mollard, <a href="http://dx.doi.org/10.1007/s00026-014-0233-x">Asymptotic Properties of Fibonacci Cubes and Lucas Cubes</a>, Annals of Combinatorics, 18, 2014, 447-457.
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,4,-2,-6,0,3,1).
%F a(n) = n*F(n+1) + (-1)^n*(n - floor(n/2)), where F(n) = A000045(n) are the Fibonacci numbers; see the formula for e'_n on p. 450 of the Klavzar - Mollard reference.
%F G.f.: z^2*(5 + 2*z - 5*z^2 - 3*z^3)/((1 + z)*(1 - z^2)*(1 - z - z^2)^2).
%e a(2) = 5; indeed Lambda(2) is the path on 3 vertices with eccentricities 2, 1, 2.
%e a(3) = 7; indeed Lambda(3) is the star on 4 vertices with eccentricities 1, 2, 2, 2.
%p with(combinat): a := n -> n*fibonacci(n+1) + (-1)^n*(n-floor(n/2)); seq(a(n), n = 0 .. 40);
%Y Cf. A000045, A210572.
%K nonn,easy
%O 0,3
%A _Emeric Deutsch_, Oct 01 2014