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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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

a(n) is the sum of the entries of row n of the triangle A235114.

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).

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

EXAMPLE

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}.

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 *)

PROG

(PARI) Vec(x*(5 - 21*x) / ((1 - 4*x)*(1 - 5*x)) + O(x^30)) \\ Colin Barker, Jul 31 2017

(MAGMA) [4*5^(n-1)+2^(2*n-2): n in [1..25]]; // Vincenzo Librandi, Aug 01 2017

CROSSREFS

Cf. A235118.

Sequence in context: A086347 A200739 A026707 * A110190 A026784 A017977

Adjacent sequences:  A235112 A235113 A235114 * A235116 A235117 A235118

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, Jan 13 2014

STATUS

approved

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Last modified November 20 13:14 EST 2019. Contains 329336 sequences. (Running on oeis4.)