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A196054 The second Zagreb index of the rooted tree with Matula-Goebel number n.

%I A196054 #14 Jan 01 2023 09:46:42
%S A196054 0,1,4,4,8,8,9,9,12,12,12,14,14,14,16,16,14,19,16,18,18,16,19,22,20,
%T A196054 19,24,21,18,23,16,25,20,18,22,28,22,22,23,26,19,26,21,22,28,24,23,32,
%U A196054 24,27,22,26,25,34,24,30,26,23,18,32,28,20,31,36,27,27,22,24,28,30,26,39,26,28,32,30,26,31,22,36,40,23,24,36,26,26,27,30,32,38
%N A196054 The second Zagreb index of the rooted tree with Matula-Goebel number n.
%C A196054 The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g.
%C A196054 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.
%H A196054 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A196054 Ivan Gutman and Kinkar C. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
%H A196054 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A196054 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A196054 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A196054 S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, <a href="http://fulir.irb.hr/751/1/CCA_76_2003_113_124_nikolic.pdf">The Zagreb indices 30 years after</a>, Croatica Chemica Acta, 76, 2003, 113-124.
%H A196054 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196054 a(1)=0; if n=p(t) (the t-th prime), then a(n) = a(t)+b(t)+G(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+b(r)G(s)+b(s)G(r); here b(m) is the sum of the degrees of the nodes at level 1 of the rooted tree having Matula-Goebel number m and G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.
%e A196054 a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3+3*1+3*1=9).
%e A196054 a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p A196054 with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+bigomega(pi(n)) else b(r(n))+b(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+bigomega(pi(n))+1 else a(r(n))+a(s(n))+b(r(n))*bigomega(s(n))+b(s(n))*bigomega(r(n)) end if end proc: seq(a(n), n = 1 .. 90);
%Y A196054 Cf. A196052, A196053.
%K A196054 nonn
%O A196054 1,3
%A A196054 _Emeric Deutsch_, Sep 28 2011