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%I #21 Nov 12 2021 03:50:18
%S 0,1,4,4,10,10,9,9,20,20,20,18,18,18,35,16,18,31,16,32,32,35,31,28,56,
%T 31,48,29,32,50,35,25,56,32,52,44,28,28,50,46,31,46,29,52,72,48,50,40,
%U 48,75,52,46,25,64,84,42,46,50,32,67,44,56,67,36,76,76,28,48,72,70,46,59,46,44,102,42,79,68,52,62,88,50,48,62,79,46,75,71,40,92,71,67,84,72,71,54,75,65,104,96
%N The Wiener index of the rooted tree with Matula-Goebel number n.
%C The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.
%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.
%H Reinhard Zumkeller, <a href="/A196051/b196051.txt">Table of n, a(n) for n = 1..10000</a>
%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H 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 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 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 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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=0; if n = prime(t) (the t-th prime), then a(n)=a(t)+PL(t)+E(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+PL(r)E(s)+PL(s)E(r); PL(m) and E(m) denote the path length and the number of edges of the rooted tree with Matula number m (see A196047, A196050). The Maple program is based on this recursive formula.
%e a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+1+1+2+2+2=9).
%e a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2; m + m(m-1)=m^2.
%p with(numtheory): a := proc (n) local r, s, E, PL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: PL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n))+PL(pi(n)) else PL(r(n))+PL(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+PL(pi(n))+1+E(pi(n)) else a(r(n))+a(s(n))+PL(r(n))*E(s(n))+PL(s(n))*E(r(n)) end if end proc: seq(a(n), n = 1 .. 100);
%o (Haskell)
%o import Data.List (genericIndex)
%o a196051 n = genericIndex a196051_list (n - 1)
%o a196051_list = 0 : g 2 where
%o g x = y : g (x + 1) where
%o y | t > 0 = a196051 t + a196047 t + a196050 t + 1
%o | otherwise = a196051 r + a196051 s +
%o a196047 r * a196050 s + a196047 s * a196050 r
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A196047, A196050.
%Y Cf. A049084, A020639.
%Y Terminal Wiener indices: A196055, A348959.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Sep 27 2011
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