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A198334 The Balaban centric index of the rooted tree having Matula-Goebel number n. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

Table of n, a(n) for n=1..80.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

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

EXAMPLE

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.

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);

CROSSREFS

Cf. A198333

Sequence in context: A082448 A190796 A070783 * A055593 A268604 A246823

Adjacent sequences:  A198331 A198332 A198333 * A198335 A198336 A198337

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Nov 27 2011

STATUS

approved

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Last modified January 17 18:27 EST 2019. Contains 319250 sequences. (Running on oeis4.)