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A324969
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Number of unlabeled rooted identity trees with n vertices whose non-leaf terminal subtrees are all different.
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13
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1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155
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OFFSET
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1,4
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COMMENTS
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A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence counts rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.
Appears to be essentially the same as the Fibonacci sequence A000045. - R. J. Mathar, Mar 28 2019
A terminal subtree T' of a tree T is a subtree all of whose vertices except one have the same degree in T' as in T itself.
The conjecture of Mathar is true. Proof: Given a rooted identity tree T, a terminal subtree T' with more than one vertex contains at least one edge that is also a terminal subtree of T'. Thus, if T has more than one branch with more than one vertex, then it fails the additional condition since it would have at least two non-leaf terminal subtrees (namely edges) that are the same. Also, T can't have under its root more than one branch with exactly one vertex because it is an identity tree. Now we know that under the root of T is exactly one branch of the same kind as T or else it has exactly one other branch with exactly one vertex. The leads immediately to the same recurrence as A000045 the Fibonacci sequence except for n=3. (End)
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LINKS
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FORMULA
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G.f.: x*(1 - x^2) / (1 - x - x^2) = x*(1 + x/(1 - x/(1 - x/(1 + x)))).
E.g.f.: -1 + x + exp(x/2)*(cosh(sqrt(5)*x/2) - (1/sqrt(5))*sinh(sqrt(5)*x/2)). - G. C. Greubel, Oct 24 2023
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EXAMPLE
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The a(1) = 1 through a(7) = 8 trees:
o (o) ((o)) (o(o)) ((o(o))) (o(o(o))) ((o(o(o))))
(((o))) (o((o))) (((o(o)))) (o((o(o))))
((((o)))) ((o((o)))) (o(o((o))))
(o(((o)))) ((((o(o)))))
(((((o))))) (((o((o)))))
((o(((o)))))
(o((((o)))))
((((((o))))))
G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 13*x^8 + ... - Michael Somos, Nov 22 2019
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MATHEMATICA
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(* First program *)
durtid[n_]:= Join@@Table[Select[Union[Sort/@Tuples[durtid/@ptn]], UnsameQ@@#&&UnsameQ@@Cases[#, {__}, {0, Infinity}]&], {ptn, IntegerPartitions[n-1]}];
Table[Length[durtid[n]], {n, 15}]
(* Second program *)
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PROG
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(PARI) {a(n) = if( n<=1, n==1, fibonacci(n-1))}; /* Michael Somos, Nov 22 2019 */
(Magma) [1] cat [Fibonacci(n-1): n in [2..50]]; // G. C. Greubel, Oct 24 2023
(SageMath) [int(n==1) +fibonacci(n-1) for n in range(1, 51)] # G. C. Greubel, Oct 24 2023
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CROSSREFS
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The Matula-Goebel numbers of these trees are given by A324968.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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