|
|
A212631
|
|
Number of dominating subsets of the rooted tree with Matula-Goebel number n.
|
|
13
|
|
|
1, 3, 5, 5, 9, 9, 9, 9, 17, 17, 17, 15, 15, 15, 31, 17, 15, 27, 17, 29, 29, 31, 27, 27, 57, 27, 53, 25, 29, 51, 31, 33, 57, 29, 53, 45, 27, 27, 51, 53, 27, 45, 25, 53, 97, 53, 51, 51, 49, 97, 53, 45, 33, 81, 105, 45, 53, 51, 29, 87, 45, 57, 89, 65, 93, 93, 27
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
a(n) is odd (see the Brouwer-Csorba-Schrijver reference).
|
|
REFERENCES
|
A. E. Brouwer, P. Csorba, and A. Schrijver, The number of dominating sets of a finite graph is odd. Preprint available on A. E. Brouwer's homepage.
|
|
LINKS
|
|
|
FORMULA
|
In A212630 one gives the domination polynomial P(n)=P(n,x) of the rooted tree with Matula-Goebel number n. We have a(n) = P(n,1).
|
|
EXAMPLE
|
a(3)=5 because the rooted tree with Matula-Goebel number 3 is the path tree R - A - B; its dominating subsets are {A}, {R,A}, {R,B}, {A,B}, and {R,A,B}.
|
|
MAPLE
|
with(numtheory): P := proc (n) local r, s, A, B, C: r := n-> op(1, factorset(n)): s := n-> n/r(n): A := proc (n) if n = 1 then x elif bigomega(n) = 1 then x*(A(pi(n))+B(pi(n))+C(pi(n))) else A(r(n))*A(s(n))/x end if end proc: B := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then A(pi(n)) else sort(expand(B(r(n))*B(s(n))+B(r(n))*C(s(n))+B(s(n))*C(r(n)))) end if end proc: C := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then B(pi(n)) else expand(C(r(n))*C(s(n))) end if end proc: sort(expand(A(n)+B(n))) end proc: seq(subs(x = 1, P(n)), n = 1 .. 100);
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|