|
%I #12 Mar 07 2017 11:29:54
%S 0,0,2,2,4,4,6,6,6,6,6,8,8,8,8,12,8,10,12,10,10,8,10,14,10,10,12,12,
%T 10,12,8,20,10,10,12,16,14,14,12,16,10,14,12,12,14,12,12,22,14,14,12,
%U 14,20,18,12,18,16,12,10,18,16,10,16,30,14,14,14,14,14
%N The Platt index of the rooted tree with Matula-Goebel number n.
%C The Platt index (or Platt number or total edge adjacency index) of a tree is the sum of the degrees of all the edges (degree of an edge = number of edges adjacent to it). See the Todeschini-Consonni reference (p. 125). It is also equal to 2 x number of paths of length 2.
%C 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.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%D A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
%D R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
%H Reinhard Zumkeller, <a href="/A198332/b198332.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=0; if n=p(t) (the t-th prime, t>=2), then a(n)=a(t)+2G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)+2G(r)G(s); G(m) denotes the number of prime di visors of m counted with multiplicities.
%e a(7)=6 because the rooted tree with Matula-Goebel number 7 is Y, where each edge has degree 2.
%p with(numtheory): a := 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 0 elif bigomega(n) = 1 then a(pi(n))+2*bigomega(pi(n)) else a(r(n))+a(s(n))+2*bigomega(r(n))*bigomega(s(n)) end if end proc: seq(a(n), n = 1 .. 90);
%o (Haskell)
%o import Data.List (genericIndex)
%o a198332 n = genericIndex a198332_list (n - 1)
%o a198332_list = 0 : g 2 where
%o g x = y : g (x + 1) where
%o y | t > 0 = a198332 t + 2 * a001222 t
%o | otherwise = a198332 r + a198332 s + 2 * a001222 r * a001222 s
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A049084, A020639, A001222, A224458.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Nov 25 2011
|
