A248086 Sum of the eccentricities of all vertices in the Lucas cube Lambda(n).

0, 0, 5, 7, 22, 37, 81, 143, 276, 490, 895, 1578, 2802, 4894, 8547, 14797, 25560, 43919, 75267, 128525, 218930, 371920, 630465, 1066452, 1800612, 3034812, 5106881, 8580883, 14398426, 24129145, 40388085, 67527563, 112786512

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

Table of n, a(n) for n=0..32.

A. Castro and M. Mollard, The eccentricity sequences of Fibonacci and Lucas cubes, Discrete Math., 312 (2012), 1025-1037.

S. Klavzar, M. Mollard, Asymptotic Properties of Fibonacci Cubes and Lucas Cubes, Annals of Combinatorics, 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);

Crossrefs

Cf. A000045, A210572.

Sequence in context: A084164 A036498 A350193 * A076409 A294154 A260658

Adjacent sequences: A248083 A248084 A248085 * A248087 A248088 A248089

Keyword

nonn,easy

Author

Emeric Deutsch, Oct 01 2014

Status

approved