OFFSET
0,3
COMMENTS
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.
a(n) = Sum(k*A210572(n,k), k=0..n).
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..1000
Aline Castro and Michel Mollard, The eccentricity sequences of Fibonacci and Lucas cubes, Disc. Math. 312 (2012), 1025-1037.
Sandi Klavžar, Michel Mollard, Asymptotic Properties of Fibonacci Cubes and Lucas Cubes, Ann. Comb. 18, (2014), 447-457.
Index entries for linear recurrences with constant coefficients, signature (1,4,-2,-6,0,3,1).
FORMULA
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.
G.f.: z^2*(5 + 2*z - 5*z^2 - 3*z^3)/((1 + z)*(1 - z^2)*(1 - z - z^2)^2).
EXAMPLE
a(2) = 5; indeed Lambda(2) is the path on 3 vertices with eccentricities 2, 1, 2.
a(3) = 7; indeed Lambda(3) is the star on 4 vertices with eccentricities 1, 2, 2, 2.
MAPLE
with(combinat): a := n -> n*fibonacci(n+1) + (-1)^n*(n-floor(n/2)); seq(a(n), n = 0 .. 40);
MATHEMATICA
A248086[n_] := n*Fibonacci[n+1] + (-1)^n*(n - Quotient[n, 2]);
Array[A248086, 50, 0] (* Paolo Xausa, Jun 09 2026 *)
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
Emeric Deutsch, Oct 01 2014
EXTENSIONS
More terms from Paolo Xausa, Jun 09 2026
STATUS
approved