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A358377
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Numbers k such that the k-th standard ordered rooted tree is a generalized Bethe tree (counted by A003238).
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17
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1, 2, 3, 4, 5, 8, 9, 11, 16, 17, 32, 37, 43, 64, 128, 129, 137, 171, 256, 257, 293, 512, 529, 683, 1024, 1025, 2048, 2185, 2341, 2731, 4096, 8192, 10923, 16384, 16913, 18725, 32768, 32769, 32897, 34953, 43691, 65536, 65537, 131072, 131329, 149797, 174763
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OFFSET
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1,2
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COMMENTS
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We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.
A generalized Bethe tree is an unlabeled rooted tree where all branches directly under the same root are equal.
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LINKS
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EXAMPLE
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The terms together with their corresponding ordered rooted trees begin:
1: o
2: (o)
3: ((o))
4: (oo)
5: (((o)))
8: (ooo)
9: ((oo))
11: ((o)(o))
16: (oooo)
17: ((((o))))
32: (ooooo)
37: (((o))((o)))
43: ((o)(o)(o))
64: (oooooo)
128: (ooooooo)
129: ((ooo))
137: ((oo)(oo))
171: ((o)(o)(o)(o))
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MATHEMATICA
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stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
srt[n_]:=If[n==1, {}, srt/@stc[n-1]];
Select[Range[1000], FreeQ[srt[#], _[__]?(!SameQ@@#&)]&]
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CROSSREFS
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These trees are counted by A003238.
A358371 and A358372 count leaves and nodes in standard ordered rooted trees.
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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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