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 A196054 The second Zagreb index of the rooted tree with Matula-Goebel number n. 2
 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, 19, 24, 21, 18, 23, 16, 25, 20, 18, 22, 28, 22, 22, 23, 26, 19, 26, 21, 22, 28, 24, 23, 32, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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. 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. REFERENCES I. Gutman and K. C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92. F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143. I. Gutman and K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92. I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142. I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22. D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273. S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, The Zagreb indices 30 years after, Croatica Chemica Acta, 76, 2003, 113-124. LINKS FORMULA 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. EXAMPLE a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3+3*1+3*1=9). a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges. MAPLE 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); CROSSREFS Cf. A196052, A196053. Sequence in context: A246066 A076359 A105675 * A292135 A053249 A071339 Adjacent sequences:  A196051 A196052 A196053 * A196055 A196056 A196057 KEYWORD nonn AUTHOR Emeric Deutsch, Sep 28 2011 STATUS approved

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Last modified April 14 06:30 EDT 2021. Contains 342946 sequences. (Running on oeis4.)