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A196053
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The first Zagreb index of the rooted tree with Matula-Goebel number n.
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3
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0, 2, 6, 6, 10, 10, 12, 12, 14, 14, 14, 16, 16, 16, 18, 20, 16, 20, 20, 20, 20, 18, 20, 24, 22, 20, 24, 22, 20, 24, 18, 30, 22, 20, 24, 28, 24, 24, 24, 28, 20, 26, 22, 24, 28, 24, 24, 34, 26, 28, 24, 26, 30, 32, 26, 30, 28, 24, 20, 32, 28, 22, 30, 42, 28, 28, 24, 26, 28, 30, 28, 38, 26, 28, 32, 30, 28, 30, 24, 38, 36, 24, 24, 34, 28, 26, 28, 32, 34, 36, 30, 30, 26, 28, 32, 46, 28, 32, 32, 36
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OFFSET
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1,2
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COMMENTS
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The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.
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.
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LINKS
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FORMULA
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a(1)=0; if n = p(t) (the t-th prime), then a(n)=a(t)+2+2G(t); if n=rs (r,s>=2), then a(n)=a(r)+a(s)-G(r)^2-G(s)^2+G(n)^2; G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.
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EXAMPLE
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a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+9+1+1=12).
a(2^m) = m(m+1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
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MAPLE
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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+2*bigomega(pi(n)) else a(r(n))+a(s(n))-bigomega(r(n))^2-bigomega(s(n))^2+bigomega(n)^2 end if end proc: seq(a(n), n = 1 .. 100);
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PROG
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(Haskell)
import Data.List (genericIndex)
a196053 n = genericIndex a196053_list (n - 1)
a196053_list = 0 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a196053 t + 2 + 2 * a001222 t
| otherwise = a196053 r + a196053 s -
a001222 r ^ 2 - a001222 s ^ 2 + a001222 x ^ 2
where t = a049084 x; r = a020639 x; s = x `div` r
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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