A198334 The Balaban centric index of the rooted tree having Matula-Goebel number n.

1, 4, 5, 5, 8, 8, 10, 10, 9, 9, 9, 13, 13, 13, 12, 17, 13, 14, 17, 14, 14, 12, 14, 20, 13, 14, 19, 20, 14, 17, 12, 26, 13, 14, 17, 21, 20, 20, 17, 21, 14, 21, 20, 17, 22, 19, 17, 29, 21, 18, 17, 21, 26, 26, 16, 29, 21, 17, 14, 24, 21, 13, 26, 37, 18, 18, 20, 21, 22, 24, 21, 30, 21, 21, 23, 29, 18, 24, 17, 30

Offset: 1

Author

Emeric Deutsch, Nov 27 2011

Keywords

• nonn

Comments

The Balaban centric index of a tree is the sum of the squares of the components of the pruning partition of the tree. The pruning partition of a tree is the reverse sequence of the number of vertices of degree 1, deleted at the successive prunings. By pruning we mean the deletion of vertices of degree 1 and of their incident edges. See the Balaban reference (p. 360) and/or the Todeschini-Consonni reference (p. 42).

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.

Examples

			a(7)=10 because the rooted tree with Matula-Goebel number 7 is Y; it has 3 vertices of degree 1 and after the first pruning we obtain the 1-vertex tree. Thus, the pruning partition is [1,3] and 1^2 + 3^2 = 10.
		

References

• A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• 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.
• R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

Links

• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A198333

Programs

Maple

with(numtheory): N := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: c := 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 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: MS := proc (m) local A, i: A[m, 1] := m: for i from 2 while 1 < A[m, i-1] do A[m, i] := c(A[m, i-1]) end do: if A[m, i-2] = 2 then [seq(A[m, j], j = 1 .. i-2)] else [seq(A[m, j], j = 1 .. i-1)] end if end proc: PP := proc (n) local NVP, q: q := nops(MS(n)): NVP := map(N, MS(n)): [NVP[q], seq(NVP[q-j]-NVP[q-j+1], j = 1 .. nops(NVP)-1)] end proc: a := proc (n) options operator, arrow: add(PP(n)[k]^2, k = 1 .. nops(PP(n))) end proc: seq(a(n), n = 1 .. 80);

Formula

A198334(n) = sum of the squares of the components of A198333(n).