A235115 Number of independent vertex subsets of the graph obtained by attaching two pendant edges to each vertex of the star graph S_n (having n vertices; see A235114).
5, 24, 116, 564, 2756, 13524, 66596, 328884, 1628036, 8074644, 40111076, 199506804, 993339716, 4949921364, 24682497956, 123144054324, 614646529796, 3068937681684, 15327508539236, 76568823219444, 382569238190276, 1911746679323604, 9554335350106916, 47754084564490164, 238700054078273156
Offset: 1
Examples
a(1)=5; indeed, S_1 is the one-vertex graph and after attaching two pendant vertices we obtain the path graph ABC; the independent vertex subsets are: empty, {A}, {B}, {C}, and {A,C}.
Links
- Colin Barker, Table of n, a(n) for n = 1..1000
- E. Mandrescu, Unimodality of some independence polynomials via their palindromicity, Australasian J. of Combinatorics, 53, 2012, 77-82.
- D. Stevanovic, Graphs with palindromic independence polynomial, Graph Theory Notes of New York, 34, 1998, 31-36.
- Index entries for linear recurrences with constant coefficients, signature (9,-20).
Crossrefs
Cf. A235118.
Programs
-
Magma
[4*5^(n-1)+2^(2*n-2): n in [1..25]]; // Vincenzo Librandi, Aug 01 2017
-
Maple
seq(4*5^(n-1)+2^(2*n-2), n = 1 .. 27);
-
Mathematica
Rest@ CoefficientList[Series[x (5 - 21 x)/((1 - 4 x) (1 - 5 x)), {x, 0, 25}], x] (* or *) LinearRecurrence[{9, -20}, {5, 24}, 25] (* Michael De Vlieger, Jul 31 2017 *) -
PARI
Vec(x*(5 - 21*x) / ((1 - 4*x)*(1 - 5*x)) + O(x^30)) \\ Colin Barker, Jul 31 2017
Formula
a(n) = 4*5^(n-1) + 2^(2*n-2) for n>=1.
G.f.: x*(5 - 21*x)/((1 - 4*x)*(1 - 5*x)).
a(n) = 9*a(n-1) - 20*a(n-2) for n>1. - Colin Barker, Jul 31 2017
Comments